Topological phase transitions in superradiance lattices

Topological phase transitions in superradiance lattices

Da-Wei Wang cuhkwdw@gmail.com Department of Physics and Centre for Quantum Coherence, The Chinese University of Hong Kong, Hong Kong, China Institute for Quantum Science and Engineering and Department of Physics and Astronomy, Texas AM University, College Station, TX 77843, USA    Han Cai Institute for Quantum Science and Engineering and Department of Physics and Astronomy, Texas AM University, College Station, TX 77843, USA    Luqi Yuan Institute for Quantum Science and Engineering and Department of Physics and Astronomy, Texas AM University, College Station, TX 77843, USA Department of Electrical Engineering, and Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA    Shi-Yao Zhu Beijing Computational Science Research Centre, Beijing 100084, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China    Ren-Bao Liu rbliu@phy.cuhk.edu.hk Department of Physics and Centre for Quantum Coherence, The Chinese University of Hong Kong, Hong Kong, China
July 5, 2019
Abstract

Topological phases of matters are of fundamental interest and have promising applications. Fascinating topological properties of light have been unveiled in classical optical materials. However, the manifestation of topological physics in quantum optics has not been discovered. Here we study the topological phases in a two-dimensional momentum-space superradiance lattice composed of timed Dicke states (TDS) in electromagnetically induced transparency (EIT). By periodically modulating the three EIT coupling fields, we can create a Haldane model with in-situ tunable topological properties, which manifest themselves in the contrast between diffraction signals emitted by superradiant TDS. The topological superradiance lattices provide a controllable platform for simulating exotic phenomena in condensed matter physics and offer a basis of topological quantum optics and novel photonic devices.

The quantum Hall effect (QHE) Klitzing1980 () reveals a new class of matter phases, topological insulators (TIs), which have been extensively studied in solid-state materials Haldane1988 (); Kane2005 (); Bernevig2006 (); Konig2007 (); Chang2013 (); Qi2009 () and recently in photonic structures Haldane2008 (); HafeziM2013 (); Rechtsman2013 (); Khanikaev2013 (); Nalitov2015 (), time-periodic systems Oka2009 (); Inoue2010 (); Kitagawa2010 (); Lindner2011 () and optical lattices of cold atoms Jotzu2014 (). The first TI is the Haldane model Haldane1988 () proposed in 1988, which shows that the QHE is an intrinsic topological property of the energy bands. It inspired the discovery of the quantum spin Hall effect Kane2005 (); Bernevig2006 (); Konig2007 () and topological superconductors Qi2009 (). The Haldane model consists of a honeycomb tight-binding lattice with complex next-nearest-neighbour (NNN) hopping, which breaks the time-reversal symmetry and induces an energy gap between two bands that have opposite Chern numbers. Inversion symmetry breaking by an on-site potential can also open band gaps. The interplay of these two types of symmetry breaking leads to transitions between phases with Chern numbers 0 and .

Notwithstanding its foundational role in topological condensed matter physics, the Haldane model has never been realized in solid-state systems. Floquet modulation of a circular polarised light in graphene was proposed to induce the complex NNN tunnelling Oka2009 (); Inoue2010 (); Kitagawa2010 (). However, the required light is soft X-ray which would directly excite electrons and be absorbed. The Haldane model of cold atoms in optical lattices Jotzu2014 () were recently realised in experiments.

Figure 1: Realisation of the Haldane model in a superradiance lattice of timed Dicke States in electromagnetically induced transparency. (a) Schematic configuration of the probe field , EIT coupling fields along and diffraction fields along and . (b) The energy level diagram of the EIT coupling, probe and scattering fields. (c) Honeycomb structure of the SL. and correspond to the two sublattices. The NNN hopping (red solid arrow), (blue dot arrow) and (green dash dot arrow) enclose a nonzero effective magnetic flux in momentum space. The arrow on the circle denotes the direction of excitation current when the Chern number .

Our recent study shows that Scully’s timed Dicke states (TDS) Scully2006 () can form a superradiance lattice (SL) in momentum space Wang2015 (). Here we propose a quantum optics realisation of an in-situ tunable Haldane model using two-dimensional SLs in a simple electromagnetically induced transparency (EIT) configuration Boller1991 (). Since Dicke’s seminal paper in 1954 Dicke1954 (), superradiance has been an important topic in quantum optics. A single photon with wave vector can excite a spatially extended -atom ensemble from the ground state to the TDS Scully2006 ()

(1)

where and are the excited and ground states of the th atom at position , respectively. The TDS stores a light momentum via phase correlations between atoms excited at different positions. This momentum can be transferred back to a single photon via directional emission Scully2006 (). By coupling to another state with three coherent plane wave fields, we construct a honeycomb SL of TDS in momentum space Wang2015 (), as shown in Fig.1. The SL Hamiltonian with rotating-wave approximation is (see Supplementary Information)

(2)

where with integers and , is the Rabi frequency of the coupling field along , are defined the same as in Eq.(1) but for states, and is the detuning of the transition frequency between and from the angular frequency of the EIT coupling fields. The states and form two sublattices of a honeycomb structure in momentum space [see Fig. 1(c)]. causes the nearest-neighbour hopping. As we will show below, periodically modulating can introduce NNN hopping with controllable phases. On the other hand, the detuning causes an energy offset between the two sublattices, breaking the inversion symmetry. Thus three Haldane phases with Chern numbers and can be in-situ realised.

Figure 2: Topological phases of the superradiance lattice. (a) The value of (defined in Eq.(5)) as a function of the modulation phases of the coupling fields and . . The signs “” are for the Chern numbers . The dashed line separates the two topological phases. (b) The two Chern numbers corresponding to two topologically distinctive configurations of the three phase angles , and on a unit circle.

We first show how to induce complex NNN hopping. We consider periodically modulated Rabi frequencies of coupling fields,

(3)

where and are the static and dynamic components of the Rabi frequency, is the modulation frequency and is the modulation phase. Here we have assumed the size of the atomic ensemble be much smaller than (where is the speed of light) and hence neglected the position dependence of the modulation. On the other hand, the ensemble size is much larger than and the number of atoms such that the TDS in the SL are approximately orthogonal to each other, i.e., . The periodic modulation induces Floquet sidebands Shirley1965 () with energy separation . While is the intra-sideband nearest-neighbour hopping, induces the inter-sideband hopping. We choose so that the Floquet sidebands are well separated in energy. Thus by second-order perturbation, the effective Hamiltonian of the NNN intra-sideband transition mediated by is (see Supplementary Information)

(4)

where the NNN hopping coefficient with . The fact that the NNN hopping coefficient is purely imaginary is crucial to the topological phases. The loop transitions via NNN hopping accumulate nonzero phases, as shown in Fig.1 (c). opens a band gap where is a dimensionless quantity determined by the summation of ,

(5)

The Chern numbers of the upper and lower bands and are (see Supplementary Information). In Fig. S.2 (a), we plot with and . The topological property of this SL Haldane model can be represented by the distribution of on a unit circle. There are two distinct topological configurations, counter-clockwise , and for , and clockwise , and for , as shown in Fig.S.2 (b). The time reversal in Eq.(S.4) is equivalent to , which leads to .

Figure 3: Topological phase transitions signatured by superradiance contrast. (a) The contrast (defined in equation (6)) as a function of the modulation phases and with . The red dot lines are zero points. . . (b) The topological phase transition between and . . 0.01 (red), 0.1 (green), 0.5 (blue) and 1.0 (black). (c) The SL band structures in the topological phase transition. . and are the two valley points. For (b) and (c), , and . For all three figures, , and .

Unlike TIs Konig2007 (); Chang2013 (), topological superradiance lattices (TSLs) have no outer edges in the semiclassical limit of the coupling fields (see Supplementary Information). Neither do TSLs have Fermi surfaces. Nonetheless, the TSL has its unique topological properties that are observable. The TDS have directional superradiance emission. Of all the TDS in the SL, only those with within the energy bands ( is the transition frequency between and ) can satisfy both energy and momentum conservation, and have directional emission in Scully2006 (). We call these states superradiant TDS and the other ones subradiant TDS. We can regard these superradiant TDS as an inner edge of a honeycomb lattice of subradiant TDS Lumer2013 (). The topological orders lead to different light emissions from different superradiant TDS. Alternatively, we can also tune the probe field frequency to test the topological band properties at certain energy, which is analogue to tuning the Fermi surface in a fermionic system.

For the sake of simplicity, we set the wavevectors of the three EIT coupling fields to be , and , and the probe field wavevector . In this case we have only three superradiant TDS with wavevectors , and , and the diffraction fields are along , as shown in Fig.1 (a). We set all fields on resonance, i.e., and . The excitation flows to and emits photons along . We denote the steady state probability amplitudes of states as and define the superradiance contrast

(6)

For , the excitation current flows along Lumer2013 (), as shown in Fig.1 (c). Since each TDS or has decoherence rate or , respectively, the excitation decays while flowing and it is more probable in state than in state . We therefore have . Similarly, for , . Thus the sign change of the superradiance contrast signatures the topological phase transition, as seen in Fig.3 (a). The superradiance contrast in Fig.3 (a) is consistent with Fig.S.2 (b) except for the two diagonal corners where the topological currents are weak and the local effect inside a unit cell dominates (see Supplementary Information).

Figure 4: Real-space propagation of a light pulse through an atom ensemble driven into a trivial or topological superradiance lattices. Diffraction of a weak Gaussian probe pulse in x-y plane for (a) a trivial SL with and (b) a TSL with . , , and . The derived linear and nonlinear susceptibilities are , in (a) and , , in (b). The parameters are such that , , and . The square denotes the area occupied by atoms. The three figures in each group are for scaled time successively. We assume the group velocity be uniform everywhere.

The topological phase with for non-zero can be reached by breaking the inversion symmetry. Substantially easier than in graphene electrons, a sublattice offset in SLs can be introduced by choosing . We set and thus when . The energy gap in absence of sublattice offset is . Topological phase transitions occur at Haldane1988 () (see Supplementary Information). In Fig.3 (b), we plot as a function of . For , the phase transition is smeared out by the relatively large . The phase transition gets more apparent for larger . Depending on , has peaks or kinks near the phase transition point because the two bands touch at the middle of the band gap, where the probe field probes, as shown in Fig.3 (c).

The TSLs have unique features in transient light propagation under pulse probe. In Fig.4, we compare the pulse propagation in a trivial SL with zero and in a TSL. For a weak probe pulse, the linear susceptibility is and the linear absorption is . The two multi-wave-mixing signals along correspond to the nonlinear susceptibilities and can be understood as a result of optical grating Wang2015 (). We simulate the pulse propagation for the three modes along using coupled wave equations Wang2013 () (see Supplementary Information). For a trivial SL without modulation, the light propagating along is symmetric, while for a TSL with , the topological currents drive the probe pulse to , even if the NNN hopping is two orders of magnitude smaller than the nearest-neighbour hopping.

TSLs can be readily realised in experiments for cold alkali atoms. Taking Rb D1 line for example, we can have , and . MHz and is controllable via inhomogeneous magnetic field Tiwari2010 (). The Rabi frequency MHz (intensity 25mW/cm). The modulation frequency can be MHz which is large enough to separate the Floquet bands. One can trap atoms in 1 mm such that and the size of the ensemble . In the K regime, the thermal random motions have negligible Doppler shifts (kHz). Another possible type of physical systems are rare earth atoms doped in solids Thiel2011 (). One should first optically pump nearly all population to , then turn on three optical fields coupling to , send in a weak field probing the to transition, and detect the diffraction signals.

Similar to the electronic TIs Lindner2011 (); Kitagawa2010 () and the optical lattice simulations of TIs Jotzu2014 (), the topological properties of TSLs are determined by Schrödinger equations, different from the photonic TIs Haldane2008 (); HafeziM2013 (); Rechtsman2013 (); Khanikaev2013 (), which are governed by Maxwell equations. The generation and detection of the topological properties of TSLs, however, can be easily controlled by light. The unique feature of the TSL is that its lattice sites have discrete momenta rather than positions. It has the advantage to be extended to dimensions higher than three where no real space lattices exist Wang2015 () and offers a platform for high-dimensional topological physics Zhang2001 ().

Funding Information

Science Foundation Grants No. PHY-1241032 (INSPIRE CREATIV) and PHY-1068554; Robert A. Welch Foundation (Grant No. A-1261); Herman F. Heep and Minnie Belle Heep Texas A&M University Endowed Fund; Hong Kong RGC/GRF; CUHK VC’s One-Off Discretionary Fund; Natural Science Foundation of China U1330203.

The authors thank Marlan O. Scully for helpful discussions.

References

  • (1) K. von Klitzing, G. Dorda and M. Pepper, Physical Review Letters 45, 494–497 (1980).
  • (2) F. D. M. Haldane, Physical Review Letters 61, 2015–2018 (1988).
  • (3) C. L. Kane and E. J. Mele, Physical Review Letters 95, 226801 (2005).
  • (4) B. A. Bernevig and S. C. Zhang, Physical Review Letters 96, 106802 (2006).
  • (5) M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766–770 (2007).
  • (6) C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, J.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, Q.-K. Xue, Science 340, 167–170 (2013).
  • (7) X.-L. Qi, T. L. Hughes, S. Raghu, S.-C. Zhang, Physical Review Letters 102, 187001 (2009).
  • (8) F. D. M. Haldane and S. Raghu, Physical Review Letters 100, 013904 (2008).
  • (9) M. Hafezi, S. Mittal, J. Fan, A. Migdall and J. M. Taylor, Nature Photonics 7, 1001–1005 (2013).
  • (10) M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev and A. Szameit, Nature 496, 196–200 (2013).
  • (11) A. B. Khanikaev, M. S. Hossein, W.-K. Tse, M. Kargarian, A. H. MacDonald and G. Shvets, Nature Materials 12, 233–239 (2013).
  • (12) A. V. Nalitov, G. Malpuech, H. Terças and D. D. Solnyshkov, Physical Review Letters 114, 026803 (2015).
  • (13) T. Oka and H. Aoki, Physical Review B 79, 081406 (2009).
  • (14) J.-I. Inoue and A. Tanaka, Physical Review Letters 105, 017401 (2010).
  • (15) T. Kitagawa, E. Berg, M. Rudner and E. Demler, Physical Review B 82, 235114 (2010).
  • (16) N. H. Lindner, G. Refael, and V. Galitski, Nature Physics 7, 490–495 (2011).
  • (17) G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif and T. Esslinger, Nature 515, 237–240 (2014).
  • (18) M. O. Scully, E. S. Fry, C. H. R. Ooi and K. Wódkiewicz, Physical Review Letters 96, 010501 (2006).
  • (19) D.-W. Wang, R.-B. Liu, S.-Y. Zhu and M. O. Scully, Physical Review Letters 114, 043602 (2015).
  • (20) K.-J. Boller, A. Imamoǧlu and S. E. Harris, Physical Review Letters 66, 2593–2596 (1991).
  • (21) R. H. Dicke, Physical Review 93, 99–110 (1954).
  • (22) J. H. Shirley, Physical Review 138, B979–B987 (1965).
  • (23) Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, Physical Review Letters 111, 243905 (2013).
  • (24) D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers and S.-Y. Zhu, Physical Review Letters 110, 093901 (2013).
  • (25) V. B. Tiwari, S. Singh, H. S. Rawat, M. P. Singh and S. C. Mehendale, Journal of Physics B 43, 095503 (2010).
  • (26) C. W. Thiel, T. Böttger, and R. L. Cone, Journal of Luminescence 131, 353–361 (2011).
  • (27) S.-C. Zhang and J. Hu, Science 294, 823–828 (2001).

Supplementary I

Appendix A Effective Hamiltonian

The three-level atoms we use to construct the two-dimensional (2D) superradiance lattice (SL) have a ground state , an excited state and a third state . The optical fields that couple and have three modes with wave vectors , and . The interaction Hamiltonian with rotating-wave approximation is

(S.1)

where and are the vacuum coupling strength and annihilation operator of the th mode, respectively. is the number of atoms and is the position of the th atom.

Figure S.1: Timed Dicke states and the associated photon numbers in an SL. The numbers associated with each TDS are the photon number difference from those of state .

Initially the atomic ensemble is in the ground state . A weak probe field with wave vector can prepare the atomic ensemble in the timed Dicke state (TDS)

(S.2)

by the collective absorption of a single photon. The combinational quantum states of the atoms and three coupling modes can be written as , where is the photon number in mode . This state is coupled to with coupling strength . The state is defined the same as in equation (S.2) with replaced by and replaced by . Similar coupling also exists for the other two modes. is in turn coupled to or via excitation by modes or . These states form a honeycomb lattice with discrete momentum coordinates, as shown in Fig. S.1, called the superradiance lattice 2014arXiv1403.7097W (). The two sublattices of the SL correspond to TDS for and . We denote the coupling field frequency as and the transition frequency between and as . The energy difference between the two sublattice is . We set the zero energy at the middle of the energies of the two sublattices. Then the energies of the and sublattices are and , respectively.

Although the coupling strengths in the SL are site-dependent, if we use coherent coupling fields with large average photon numbers , the Rabi frequency of the th mode can be approximated as the classical Rabi frequency . The SL with quantized photon numbers has edges when either of the three coupling field photon numbers reduces to zero. The total lattice has a triangular boundary. However, since the TDS have finite life time, which we suppose to be on average, once we create an excitation in the SL, the average distance this excitation can travel is in the order of . In this paper we have and the edges can never be reached. Interesting edge effects might exist in the few-photon limit and it will be discussed elsewhere. Since the photon numbers are correlated to the momenta ’s in the TDS, we can simplify the notations by dropping the photon numbers. The Hamiltonian of these TDS is

(S.3)

where with integers and .

The next-nearest-neighbour (NNN) hopping terms are induced by the periodic modulation of the three coupling field Rabi frequencies,

(S.4)

where and are the static and dynamic components of the Rabi frequencies. is the modulation frequency. is the modulation phase of the th field. is the modulation wavevector. We choose the atomic ensemble much smaller than such that the position dependence of the phase can be neglected. We set for simplicity. On the other hand, the size of the atomic ensemble shall be much larger than and the total number of atoms such that

(S.5)

We assume the atoms be randomly distributed and their number is large enough to cover all possible points in the real space Brillouin Zone. The SL can be regarded as infinite.

We expand the Hamiltonian into static, positive- and negative-frequency components,

(S.6)

where

(S.7)
(S.8)
(S.9)

Note that are not Hermitian themselves, but is the Hermitian conjugate of . The phase factors in play the crucial role for the complex NNN hopping terms in the Haldane model.

The dynamics of the system is a Floquet problem Shirley1965 (); Kitagawa2011 (). Based on the Floquet theorem, the wave function can be written as

(S.10)

where is the quasi-eigenenergy and

(S.11)

is a periodic wave function with period . Substituting equations (S.6)-(S.11) to the Schrödinger equation, , we obtain

(S.12)

The terms with the same time evolution phase factors should be equal on both sides. We therefore have

(S.13)

The quasi-eigenenergy can be obtained by diagonalizing the above Hamiltonian.

For the sake of simplicity, we assume the separation between the Floquet sidebands is much larger than the bandwidth, , where the perturbation theory can be applied Liu2000 (); Gomez-Leon2013 (). When the probe field is near resonance, where is the transition frequency between and and is the probe field frequency, only the Floquet band with in Eq.(S.13) is relevant. For states with eigenfrequencies near , the effective Hamiltonian can be obtained by standard second-order perturbation as

(S.14)

where

(S.15)

with NNN Hamiltonian

(S.16)

Explicitly,

(S.17)

where with . The crucial factor comes from the quantum interference between the two pathways shown in Fig.S.2 (a). The loop transitions via NNN hopping accumulate nonzero phases due to these complex factors, as shown in Fig.S.2 (b).

Figure S.2: Complex next-nearest-neighbor hopping induced by Rabi frequency modulation. (a) The Floquet sidebands of the SL and the NNN transition (dashed arrow) induced by the interference between two second-order inter-sideband transitions (solid arrow). Phases are labelled for each step. (b) The effective NNN transitions in a unit cell. The NNN hopping (red arrow), (blue dot arrow) and (green dash dot arrow) enclose a nonzero effective magnetic flux.

The effective Hamiltonian is greatly simplified in the real-space representation. We denote the real-space basis states as

(S.18)

The effective Hamiltonian can be written as

(S.19)

where the effective magnetic field with

(S.20)
(S.21)
(S.22)

and the pseudo spin with the Pauli matrices for the th atom defined as , and .

Appendix B Berry connection and Berry curvature

The topological properties of the wavefunctions are determined by the Berry connection and Berry curvature ,

(S.23)
(S.24)

where and are the gradient and curl operators with respect to .

Figure S.3: Effective magnetic fields in a Brillouin zone for topological and trivial SL’s. (a) A topological SL with . (b) A trivial SL with . The arrows show the vector and the colours show . The and axis are in unit of . The phase of the coupling field is and . Note that the points at the boundaries of the first Brillouin zone are equivalent.

Similar to a spin-1/2 in a magnetic field , the eigenenergies of the SL eigenstates of the effective Hamiltonian in equation (S.19) are

(S.25)

where are for the upper and lower bands and is the magnitude of . The eigen wavefunction in the upper band can be written as

(S.26)

This wavefunction is well defined except for the south pole of the Bloch sphere where with . The eigen wavefunction near the south pole can be written as

(S.27)

where the gauge transformation

(S.28)

This way, the whole Bloch sphere is fully covered by two gauges Wu1975 ().

The Berry connection in the upper band (except for the south pole of the Bloch sphere) is

(S.29)

and near the south pole we can use

(S.30)

The Berry curvature is

(S.31)

where with is the Levi-Civita symbol. The Chern number is defined as the total Berry curvature in the whole first Brillouin zone

(S.32)

which counts the winding number of the effective magnetic field wrapping around the Bloch sphere in the whole Brillouin zone Qi2006 (); Katan2013 ().

In Fig.S.3, we plot in the Brillouin zone for a topological nontrivial SL where the band gap is opened by the NNN hopping. If we go from the point to the point , moves from the north pole to the south pole of the Bloch sphere. The Chern number is one. If the band gap is opened by the on-site offset, , is a constant in the Brillouin zone. can only cover a patch on the Bloch sphere and the Chern number is zero.

The two bands of have the smallest gap at the and points , where

(S.33)

and hence . At these symmetry points of the Brillouin zone the Hamiltonian is diagonal,

(S.34)

where .

Figure S.4: The eigenstates at and points for topological and trivial SL’s. (a) For a topological SL, the eigenstate at (i.e., point) is in the upper band while it is in the lower band at (i.e., point). (b) For a trivial SL, is the eigenstate in the upper band at both and . Red (Blue) lines denote the bands with more components in ().

Let us consider the specific case that , , and thus . In this case has opposite signs at , as shown in Fig.S.3 (a). It is obvious that at , the eigenstate in the upperband with eigenenergy is , while at , is the eigenstate in the lower band, as shown in Fig.S.4 (a).

Near , we can use equation (S.26) to describe the wavefunction. However, equation (S.26) cannot describe all the wavefunctions in the upper band. It has a singularity at where the magnetic field points to the south pole. We can remove the singularity by tuning . When , the effective magnetic field does not experience the whole Bloch sphere [see Fig.S.3 (b)]. In this case, in equation (S.34) has the same sign for , is in the upper band at both , as shown in Fig.S.4 (b). The wavefunction can be described by the same gauge. Thus the single-valued Berry connection has no singularities on a closed surface and the Chern number is zero. Generally, the Chern number of the upper band can be written as

(S.35)

Specifically, when , .

Appendix C Dynamic evolution

We can solve the dynamics of the atomic ensemble in real space with the total Hamiltonian including the probe field,

(S.36)

with being the probe detuning with respect to the middle of the band gap and the probe field Rabi frequency. The wavefunction in real space

(S.37)

The dynamic equations of the probability amplitudes are

(S.38)
(S.39)

where is the decoherence rate of states.

In the limit of weak probe field , we have . In the steady state, . From Eq.(S.39), we obtain

(S.40)

Substituting Eq.(S.40) in Eq.(S.38), we obtain

(S.41)

In the SL coordinates, the wavefunction can be written as

(S.42)

where the probability amplitude

(S.43)

For uniformly distributed atoms, in the limit , we can assume all points in the Brillouin zone are occupied by atoms, so the summation can be written as integration in the first Brillouin zone,

(S.44)

where is the area of the first Brillouin zone.

Appendix D Diffraction Contrast and topology

In this section, we quantitatively compare the contrast between and in topological and trivial SL’s. We assume and . Then Eq.(S.41) becomes

(S.45)

We assume and thus is highly centred at and points, where we have and

(S.46)

According to Eq.(S.44), the probability amplitude can be approximately calculated by the integration of in small areas near ,