Topological physics has recently attracted great attention due to its fundamental scientific importance and potential application in novel optoelectronic devices [1–6]. Silicene, which consists of silicon atoms [7,8], with a monolayer honeycomb system is a prime candidate. Unlike graphene, silicene is nonplanar and possesses intrinsic spin-orbit coupling that provides a mass to Dirac electrons; it also favors an experimentally accessible Kane–Mele type quantum spin Hall effect [9,10]. It is remarkable that the mass can be controlled by an external electric field and/or circularly polarized light [11,12], which leads to topological phase transitions in silicene. The Chern number, as a topological invariant, is associated with a band, and it is the integral of the Berry curvature over the entire Brillouin zone [13]. Further, Berry curvature is a function of wave vectors. Measurement of the Chern number plays an important role in the determination of different topological phases. However, one of the most challenging goals is how to measure the Chern number and different topological states in a direct and facile way.

Here, considering that Hall conductivity can be directly reflected by an optical signature, we investigate the signature amplified by weak measurement [14]. An in-plane photonic spin Hall effect (SHE) and longitudinal phase shifts can be a new spin-sensitive topological probe [15]. A fundamental origin of the photonic SHE is related to the spin-orbit interaction. The ubiquitous effect is often used as a metrological tool [16–20] because of its high sensitivity to the physical properties of materials. This provides a novel measuring method for the metrology community [21–23].

This paper presents an optical signature for probing low-energy Dirac-like physics in systems that can realize topological phase transitions. It is valuable since an optical signature will make this important physics more accessible to the metrology community. The in-plane photonic SHE serves as an optical probe by virtue of the spatial splitting and angular deviations of its reflected beam. We can observe different Hall transitions while silicene samples were exposed to externally applied circularly polarized lasers and/or static electric fields. While Ref. [24] considered transverse splitting, its system did not exhibit weak measurement. This weak value amplification technique can be applied to enhance the original tiny signature, achieving a sensitivity to ∼0.5mm (∼1 milliradian). Further, Brunner and Simon present a purely imaginary weak value that can detect phase shifts, which has potential to outperform standard interferometry by three orders of magnitude [25]. We apply this progressive method to probe topological phase transitions on 2D atomic crystals. Thus, the optical signature that is amplified by weak measurement provides a feasible scheme for observing the Chern number and topological phase transitions.

We consider optical reflection from an air–silicene interface to measure signals of topological phase transition. Due to the refractive index gradient, the momentum can be transferred between the spin and extrinsic orbital angular momentum of light along its propagation trajectory, resulting in beam splitting with polarizations [26]. The beam splitting in the photonic SHE, impinging on an anisotropic medium, occurs, respectively, in the transverse and in-plane directions [27]. The transverse splitting, implied by angular momentum conservation [24,28,29], is ubiquitous, while the in-plane splitting only occurs on an anisotropic medium [30]. The cross-polarization can cause in-plane splitting [31]. Further, the Hall conductivity gives a non-negligible contribution to the cross-polarization [32,33]. The in-plane photonic SHE is distinguished from the Goos–Hänchen effect [34]. Because the former is spin-dependent, and it takes place as a result of an effective spin-orbit interaction, the latter is polarization-dependent, which is described in terms of evanescent wave penetration. Remarkably, spin-sensitive topological phase transition requires a spin-sensitive measurement. Here, we propose to take the in-plane splitting as an optical probe of weak measurement to detect topological phase transitions. Thus, a phase estimation technique is introduced to complete the entire detection system.

Figure 1 illustrates that a Gaussian beam of frequency ω0 impinges at an angle θi on a silicene placed on top of a SiC substrate of dielectric constant ϵ. The system is subject to a static electric field and circularly polarized light. The corresponding electromagnetic potential is given by A(t)=(A0sinωt,A0cosωt), where ω≫ω0 is the frequency of circularly polarized light with ω<0(ω>0) specifying left (right) circular polarization and A0 is the amplitude.

The low-energy effective Hamiltonian is derived in the momentum space [35–37]. We have Hη=ℏνF(ηkxτx+kyτy)+λSOσzητz−eℓEzτz−Λητz.(1)Here, σ=(σx,σy,σz) and τ=(τx,τy,τz) are the Pauli matrices of the spin and the sublattice pseudospin, respectively; η=±1 is the valley index, νF is the Fermi velocity, λSO is the effective spin-orbit (SO) coupling, ℓ is the buckled height, Ez is the perpendicular electric field, and Λ is the coupling constant between the monolayer and the circularly polarized light. By diagonalizing the Hamiltonian in Eq. (1), the bandgap is given by 2|msη| with the Dirac mass: msη=ηsλSO−eℓEz−ηΛ.(2)Here, Λ=(eaA0νF)2ℏ/ω0, and a is the lattice constant. From Eq. (2), the Dirac mass can be controlled by applying electric field Ez and coherent laser beam.

In addition, the nontrivial topological feature can also be characterized by the Chern number. In the momentum space, the vector Berry connection is defined as Ai(k→)=−i⟨ψ(k→)|∂∂ki|ψ(k→)⟩, where ψ(k→) is the eigenvector of the Hamiltonian. The Berry curvature is F(k→)=∂Ay(k→)/(∂kx)−∂Ax(k→)/(∂ky). Since the different states correspond to different beam shifts, the Berry curvature can also be distinguished by photonic SHE [38,39]. It is calculated in Fig. 2(h).

The Chern number is the integral of the Berry curvature F(k→) over the first Brillouin zone. It is also associated to a band provided the gap is open. It is determined by the Dirac mass and the valley index, that is, Csη=ηsgn(msη)/2. Then, the Chern number is obtained by C=∑η=±(C+η+C−η). The Chern number C is shown in Fig. 2. In Figs. 2(a)–2(c), we show that, if we increase Λ (the initial value Λ=0) while keeping Ez=0, these gaps close at Λ=λSO, where silicene is a semimetal. The spin-polarized metal (SPM) appears at this point. It has the identical spin configuration at K and K′ points. Further increasing Λ, the gap is reopened. And the Chern number changes, indicating that a system is continuously deformed into a new state from the previous state, namely, a topological phase transition occurs. Similar transitions can appear by changing Ez in Figs. 2(a), 2(d), and 2(f). The system undergoes topological phase transitions from a quantum spin Hall insulator (QSHI) into a quantum anomalous Hall insulator (QAHI) or a band insulator (BI). These gaps will also close when |Ez| reaches a critical value. The spin-valley-polarized metal (SVPM) emerges at this point between the QSHI and BI. In Fig. 2(g), a single Dirac cone (SDC) state emerges along the phase boundaries, where the gap is open at the K point but closed at the K′ point. Further increasing Λ and Ez from QSHI, the silicene reaches the photoinduced spin-polarized quantum Hall insulator (PS-QHI) phase in Fig. 2(e).

The expressions for the longitudinal and transverse conductivities with zero temperature can be obtained by Kubo’s formula [24,40–42]; we have $$\begin{gathered} σ˜xxηsσ0/2π=4μ2−|msη|22ℏμΩΘ(2μ−|msη|)+(1−|msη|2ℏ2Ω2) \\ ×arctan(ℏΩM)+|msη|2ℏΩM, \\ σ˜xyηsσ0/2π=2ηmsηℏΩarctan(ℏΩM). \end{gathered}$$(3)Here, Ω=−iω+Γ, where Γ is the scattering rate. σ0=e2/4ℏ, σ˜xxηs=σ˜yyηs, σ˜xyηs=−σ˜yxηs, and M=max(|msη|,2|μ|).

The reflected amplitudes can be obtained by solving Maxwell’s equations and imposing the appropriate boundary conditions, Ei+Er=Et and Hi+Hr−Ht=J. We decompose the incoming field into its h and v components. Sequentially, the four Fresnel’s reflection coefficients, rij for incoming i-polarized and outgoing j-polarized waves, are obtained [43,44]: $$\begin{gathered} rhh=α−β++σhvσvhα+β++σhvσvh,rvv=α+β−−σhvσvhα+β++σhvσvh, \\ rhv=−2Ziσhvα+β++σhvσvh,rvh=2Ziσvhα+β++σhvσvh. \end{gathered}$$(4)Here, Zt=1/ϵt, α+=σhhkiz/ki+kizkt/(Ztkiktz)+1/Zi, α−=σhhkiz/ki+kizkt/(Ztkiktz)−1/Zi, β+=−σvvki/kiz−kiktz/(Ztkizkt)−1/Zi, β−=σvvki/kiz+kiktz/(Ztkizkt)−1/Zi, Zi=1/ϵi, and ki is the wavenumber.

For horizontal polarization state |H⟩ and vertical polarization state |V⟩, the reflected polarization states can be written as [|H(kr)⟩|V(kr)⟩]T=MR[|H(ki)⟩|V(ki)⟩]T. Here, MR can be expressed as (rhhrhvrvhrvv).(5)In the above equation, the boundary condition is krx=−kix and kry=kiy. By making use of Taylor series expansion based on the arbitrary angular spectrum component, all Fresnel’s reflection coefficients rij can be expanded as a polynomial of kix. Then, we have |H(ki)⟩→(rhh−krxk0∂rhh∂θi)|H(kr)⟩+(rvh−krxk0∂rvh∂θi)|V(kr)⟩,(6)|V(ki)⟩→(rhv−krxk0∂rhv∂θi)|H(kr)⟩+(rvv−krxk0∂rvv∂θi)|V(kr)⟩.(7)The polarization states of |H⟩ and |V⟩ are decomposed into two spin components |H⟩=(|+⟩+|−⟩)/2 and |V⟩=i(|−⟩−|+⟩)/2, where |+⟩ and |−⟩ represent the left- and right-circularly polarized components, respectively. The wave function in momentum space (k) is specified by the following expression: |Φ⟩=w02πexp[−w02(kix2+kiy2)4].(8)From Eqs. (6), (7), and (8), for weak spin-orbit interaction, the reflected wave functions |ΦrH⟩ and |ΦrV⟩ can be obtained: |ΦrH⟩≈rhh−irvh2exp(iskrxδxH+krxΔxH)|+⟩|Φ⟩+rhh+irvh2exp(iskrxδxH+krxΔxH)|−⟩|Φ⟩,(9)|ΦrV⟩≈rhv−irvv2exp(iskrxδxV+krxΔxV)|+⟩|Φ⟩+rhv+irvv2exp(iskrxδxV+krxΔxV)|−⟩|Φ⟩.(10)Here, the term of photonic spin Hall shifts can be written as δxH=rhhki(rhh2+rvh2)∂rvh∂θi−rvhki(rhh2+rvh2)∂rhh∂θi,(11)δxV=rhvki(rvv2+rhv2)∂rvv∂θi−rvvki(rvv2+rhv2)∂rhv∂θi.(12)The term exp(iskrxδrxH,V) is the origin of the spin-orbit interaction. In general, the phases φG=skrxδrxH,V can be regarded as the spin-redirection Berry phases [45–47]. It should be noted that the above approximations do not hold for strong spin-orbit interaction δrxH,V≈w0.

The spatial and angular optical signatures of the wave packet at the initial position (zr=0) are given by ⟨Δx±H,V⟩=⟨Φr±H,V|i∂krx|Φr±H,V⟩⟨Φr±H,V|Φr±H,V⟩,(13)⟨Θx±H,V⟩=1ki⟨Φr±H,V|krx|Φr±H,V⟩⟨Φr±H,V|Φr±H,V⟩.(14)Substituting Eqs. (9) and (10) into Eqs. (13) and (14), the in-plane photonic spin Hall shifts can be written as ⟨ΔxsH,V⟩=−sRe[δxH,V],(15)⟨ΘxsH,V⟩=−1zRsIm[δxH,V].(16)Here, zR=πw02/λ is the Rayleigh range, and w0 is the beam waist. The optical signature also manifests itself as an in-plane photonic SHE whose magnitude of spatial shifts (angular deviations) reaches the order of incident wavelength (one tenth of mrad). The effect of topological phase transition in photonic SHE is clearly influenced for infrared frequencies (in the range shown 0.44<Re[ϵ]<1.52 and Im[ϵ] < 0.09). Figure 3 shows that the topological phase diagram can be clearly described by optical signatures, whether it is spatial splitting [Figs. 3(a) and 3(b)] or angular deviations [Figs. 3(c) and 3(d)] of the reflected beam. Due to the spin-dependent nature of photonic SHE, a different Chern number can also be described.

Actually, this tiny original signature needs to be amplified. Considering its practicality, using weak measurement to amplify the signature is essential [48–50]. After the preselected state, weak interaction, and post-selected state, the wave function evolves to the final state: $$\begin{gathered} |Φf⟩=⟨ψf|exp(iσzkrxδrx)|ψi⟩|Φ⟩ \\ =⟨ψf|1+iσzkrxδrx|ψi⟩|Φ⟩ \\ ≈⟨ψf|ψi⟩(1+ikrxδrx⟨ψf|σz|ψi⟩⟨ψf|ψi⟩)|Φ⟩ \\ =⟨ψf|ψi⟩(1+ikrxAwδrx)|Φ⟩. \end{gathered}$$(17)Here, Aw is the weak value and can be written as Aw=⟨ψf|σz|ψi⟩⟨ψf|ψi⟩,(18)where ψi is the initial state, called the preselection, and ψf is the final state, called the postselection: Im[AwδrxH,V]=Re[Aw]Im[δrxH,V]+Im[Aw]Re[δrxH,V],(19)Re[AwδrxH,V]=Re[Aw]Re[δrxH,V]−Im[Aw]Im[δrxH,V].(20)Here, the amplified angular and spatial shifts are determined by the imaginary and real parts of the weak values.

To obtain a clear picture, we represent the preselection and postselection of the states on a Bloch sphere as shown in Fig. 4. The preselected state can be written as |ψi⟩=cos(Θ2)|+⟩+e−iΦsin(Θ2)|−⟩,(21)where 0≤Θ≤π and 0≤Φ≤2π represent the quantum state on the Bloch sphere [51]. Due to the polarized rotation effect, we introduce a middle polarization state: |ψm⟩=cos(Θ2+α)|+⟩+e−i(Φ+2δ)sin(Θ2+α)|−⟩.(22)

Here, α=arctan(|rhv|/|rhh|) and δ=[arg(rhv)−arg(rhh)]/2 are the polarized rotation angles for |H⟩ polarization. After the postselection, the final polarization state can be written as |ψf⟩=sin(Θ2+α+Δ1)|+⟩−e−i(Φ+2δ+2Δ2)cos(Θ2+α+Δ1)|−⟩,(23)where Δ1 and Δ2 are the postselection angles. A condition for weak measurements is that the two polarization states |ψi⟩ and |ψf⟩ are required to be perpendicular to each other [52]. Therefore, a middle polarization state is introduced, and the weak value Aw should be modified as Aw=⟨ψf|σz|ψm⟩⟨ψf|ψm⟩.(24)The probability of the postselection on ⟨ψf| and the weak value take the form γ≈|⟨ψf|ψm⟩|2=cos2Δ2sin2Δ1+sin2Δ2sin2(Θ+Δ1+2α),(25)Re[Aw]=sinΔ1sin(2α+Δ1+Θ)|⟨ψf|ψm⟩|2,(26)Im[Aw]=sin2Δ2sin(2α+Θ)sin[2(α+Δ1)+Θ]2|⟨ψf|ψm⟩|2.(27)After a free evolution, the wave packet evolves to its final position: ⟨ΔxwH,V⟩=⟨Φf|i∂krx|Φf⟩⟨Φf|Φf⟩,(28)⟨ΘxwH,V⟩=1ki⟨Φf|krx|Φf⟩⟨Φf|Φf⟩.(29)Note that the angle Δ1 (Δ2) is responsible for the real (imaginary) part of the weak value. If only purely real weak value is considered, we could make Δ2=0. Then, γ=sin2Δ1 and Aw=sin(2α+Δ1+Θ)/sinΔ1. From Eqs. (28) and (29), the amplified shifts for |H⟩ and |V⟩ polarization are given by ⟨ΔxwH,V⟩=Re[AwδrxH,V]=Re[δrxH,V]sin(2α+Δ1+Θ)sinΔ1,(30)⟨ΘxwH,V⟩=1zRIm[AwδrxH,V]=Im[δrxH,V]sin(2α+Δ1+Θ)zRsinΔ1.(31)Moreover, if only a purely imaginary weak value is considered, we could make Δ1=0. So, γ=sin2Δ2sin2(Θ+2α) and Aw=−icotΔ2. Similarly, the amplified spatial and angular shifts can be obtained.

Here, the weak value can bring the spatial shifts (angular shifts) to the order of millimeters (milliradians). Based on the angular spectrum theory, the optimum of weak measurement can be obtained [53]. In Figs. 5(a) and 5(b), the optimum points (Δ1,Δ2) of amplified spatial shifts are close to points (−2,10) and (2,−10). Further, the optimum of amplified angular shifts is close to point (2, 10) and (−2,−10), respectively. Near the optimum point, the detected spatial (angular) shifts are about 50 (10) times larger than the initial values.

The spatial intensity profiles in different states are shown in Fig. 6. Figures 6(a)–6(f) show the gain effects of weak measurement and the result of interfering destructively in weak measurement. There is a trade-off in using the weak value to amplify the tiny signal: the sacrifice lies in a loss of energy. Note that the shifts refer to the distance from the centroid of reflected field to the center of the reflection coordinate system. Figure 6(a) shows the original intensity distribution I for |H⟩ polarization in the state of QSHI. Due to the spin-orbit interaction, the unperturbed photons exhibit tiny spin-dependent splits. Subsequently, the postselection technique is used to act on the spin state of the photons, as shown in Fig. 6(d). We normalize the unperturbed intensity so that the loss of energy can be more intuitively reflected. From Figs. 6(g)–6(i), since the topological states are protected, the optical signal is stable at the low-energy region. When Λ is increased, the gap is closed; the system also becomes an intermediate state in Figs. 6(b) and 6(e), namely, the state of SPM. Simultaneously, the steady state of the signature is also broken, that is, the shifts change. Finally, Λ is further increased, the bandgap is reopened, and the topological phase transition occurs; then, the Chern number changes, as shown in Figs. 6(c) and 6(f). This means that the signature is protected by the new topological state, that is, the state of QAHI. The shift is restabilized. Comparing Figs. 6(h) with 6(i), the signature differences in different topological states are further amplified due to the weak measurement technique. This also makes it possible to observe the topological phase transition via convenient and direct optical experiment.

The weak measurement technique gives access to increase the detector’s resolution. Unlike our previous scheme of using the distribution of the light field as a probe, we use the distribution of frequency to implement the phase signal estimation. The time of arrival of the photon is considered as the probe. This scheme can be used to measure the time delay in materials, depending on the polarizations. The preselection and postselection result in an amplification of the delay. The weak value serves as an amplification factor. The frequency-domain detection in weak measurement could, in principle, outperform previous standard interferometry by three orders of magnitude [25]. Here, we use this progressive method to measure 2D atomic crystals.

We have a Gaussian function f(ω)=(πσ2)−1/4×exp[−(ω−ω0)2/2σ2] [54,55]. Since the Soleil–Babinet compensator (SBC) is responsible for weak interaction, the preselected and postselected states are different from the previous one. The preselected state is considered an elliptical polarization |ψi⟩=(e−iϵ|H⟩+e+iϵ|V⟩)/2; the postselected state is written as |ψf⟩=(|H⟩−|V⟩)/2. Thus, we have a purely imaginary weak value Aw=icotϵ. The quantum system is the photonic polarization. The fundamental physical process is this way: photons emitted from a source with a certain spectral width pass through a polarizer and a quarter-wave plate, that is, the preselection state. The photons with an elliptically polarization state are then reflected on a topological material, which is silicene. Next, the photons pass SBC and another polarizer with postselection probability. Therefore, we have $$\begin{gathered} |Φf⟩=⟨ψf|exp(−iτωσ^)Mr|ψi⟩⊗|Φ⟩ \\ =0.5f(ω)(e−iτω−iϵrhh−e+iτω−iϵrvh+e−iτω+iϵrhv−e+iτω+iϵrvv). \end{gathered}$$(32)The frequency distribution is written by F(ω)=|⟨Φf|Φf⟩|2. The center of frequency distribution is ∫ωF(ω)dω∫F(ω)dω=ω0+Δω,T=∫F(ω)dω.(33)In Fig. 7(a), the spectral width is assumed as 100 nm. Due to the decoherence, the postselection probabilities will eventually tend to 0.5. That is to say, excessive time delays are not allowed. Fortunately, the time delay provided by SBC is significantly smaller than the decoherence time. Since topological states are protected, the decoherence rate of postselection probabilities is the same in different states. However, due to the closed gaps, the situation will be different when Λ/λSO=±1.

The silicene is anisotropic topological materials; thus, due to the existence of cross-polarization, polarized rotation is inevitable. It can also lead to a large selected angle in preselection. In Fig. 7(b), the selected angle ϵ is close to 35.5 deg. Compared with the case without silicene, of course, there is no polarization rotation without silicene, we need to rotate the polarizer to compensate for this material-induced polarization rotation. So, as long as we can use SBC to compensate for the material-induced phase difference between |H⟩ and |V⟩ states, we can obtain the specific polarized rotation angle, which is caused by the material. Note that the sensitivity of weak value amplification is different in different states. When Λ/λSO=±1, the sensitivity is the same because the bandgaps are closed.

The frequency shift Δω is responsible for the spectral shift Δλ. We have Δλ=λ22πcΔω.(34)From Figs. 7(c) and 7(d), the sensitivity of spectral shifts is different in different states. However, the selected angle is different from the previous one. Because the symmetry point does not appear when ϵ=0, we can get ϵ=ϵ0+Δϵ. Further, Δϵ is the small angle we mentioned above. The ϵ0 is attributed to material-induced polarization rotation. By properly adjusting Δϵ, we can obtain the optimal weak value. In addition, the smaller spectral width or time delay will result in small phase shifts. The smaller Δϵ will be more suitable.

In conclusion, we have discussed the optical signature in a topological phase transition. The Chern number and topological phase can be determined by the in-plane photonic spin Hall effect. Furthermore, a signature-enhancement technique known as quantum weak measurement has been theoretically proposed to enhance the original signature. Then, a phase estimation technique for 2D atomic crystals has been proposed. Given our previous work for measuring the photonic spin Hall effect through weak measurement [56], an experimental demonstration of this work is within current capabilities. Because of the importance of measurement in topological materials, the introduction of weak measurement into topological phase transitions will significantly enrich the field of topological physics [57–59].