Terahertz Optics Driven Phase Transition in Two-Dimensional Multiferroics

Displacive martensitic phase transition is potentially promising in semiconductor based data storage applications with fast switching speed. In addition to traditional phase transition materials, the recently discovered two-dimensional ferroic materials are receiving lots of attention owing to their fast ferroic switching dynamics, which could tremendously boost data storage density and enhance read/write speed. In this study, we propose that a terahertz laser with an intermediate intensity and selected frequency can trigger ferroic order switching in two-dimensional multiferroics, which is a damage-free noncontacting approach. Through first-principles calculations, we theoretically and computationally investigate optically induced electronic, phononic, and mechanical responses of two experimentally fabricated multiferroic (with both ferroelastic and ferroelectric) materials, \b{eta}-GeSe and {\alpha}-SnTe monolayer. We show that the relative stability of different orientation variants can be effectively manipulated via the polarization direction of the terahertz laser, which is selectively and strongly coupled with the transverse optical phonon modes. The transition from one orientation variant to another can be barrierless, indicating ultrafast transition kinetics and the conventional nucleation-growth phase transition process can be avoidable.


Introduction.
Optical control of material geometries is receiving rapidly growing attentions in very recent years. For example, some ferroelectric/multiferroic perovskites, such as BaTiO3 1,2 , BiFeO3 3 , and SrTiO3 4,5 would experience their ferroic order switch under laser pulse illumination (with the optical frequency well-below their electronic bandgaps). It offers a remarkable and promising scheme to manipulate the structure of materials, avoiding mechanical or electrochemical contacts with these samples, which might slow down the effect and introduce unwanted impurities or disorders. Thus, this fast-paced noncontacting opto-mechanical strategy is less susceptible to lattice damage and provides an ultrafast manipulation of materials on picosecond time scales and submicrometer length scales 6 .
According to optical response theory, there are four regimes of optical frequency, namely, low frequency regime where optical frequency < − ( is energy difference between two states and is lifetime), absorption frequency regime where − < < + , reflection regime where + < < (where is plasmon frequency), and transparent regime where > . The second frequency regime may introduce significant heat, the light is highly reflected in the third frequency regime, and the optical response in transparent regime is usually small. Hence, low optical energy light would be intriguing to control the material behaviors practically 7 .
Low optical energy light can be further classified into two categories: near or midinfrared optics with its photon energy above phonon but below electronic bandgap of semiconducting materials [such as experiments on BaTiO3, Ref. (1,2)], and far infrared optics with its (terahertz) frequency strongly and directly coupled with phonons.
Atomically thin two-dimensional (2D) materials, with their extremely large surface area-to-volume ratio, are more optically addressable and accessible. Therefore, noncontacting optically driven ferroic order transition in 2D ferroic materials will be promising with their easy and damage-free manipulation, large information storage density, and ultrafast kinetics. Previous theoretical studies mainly use parameterized model including phonon-phonon nonlinear interactions 8,9 . In this work, we theoretically and computationally evaluate the interactions between light and phonons, as well as light and electrons in 2D time-reversal invariant multiferroic (with both ferroelectric and ferroelastic orders) materials, from a first-principles approach. We choose two experimentally fabricated systems, i.e., monolayers β-GeSe 10 and α-SnTe 11 , to illustrate our theory. We predict that linearly polarized terahertz laser (LPTL) pulses with intermediate intensity (around 1-2 MV/cm) can trigger ferroic order switch in these systems. A contact-free direction-dependent vibrational electron energy loss spectroscopy (EELS) is also theoretically calculated, which can be used to detect and measure the structural signature in a high resolution. In addition, we calculate the second harmonic generation (SHG) effects, which also couple to ferroic orders. These strategies could provide powerful and non-invasive tools to characterize ferroicity that is indistinguishable by the traditional diffraction methods.
We analyze the light-matter interaction thermodynamically. When LPTL is illuminated onto a semiconductor (with optical bandgap larger than ~40 meV), the electron-hole pair formation is eliminated owing to small photon energy. Therefore, only optical electromagnetic field effects need to be considered. Here we are interested in the time-reversal invariant systems, in which the magnetic field interaction is very weak and can be omitted. Hence, we focus on the alternating electric field component (ℰ ⃑ = ⃑ , ω ~ THz) effect, here is the maximum value of the electric field.
The LPTL would accelerate both electronic and phononic subsystems in the material, and its work done per volume can be evaluated by = Re ℰ ⃑ • ⃑ * , where ⃑ is time dependent polarization density. Here, closed boundary condition 12,13 is applied, since electric polarization is in the xy-plane. Using Legendre transformation, the Gibbs free energy (GFE) density change is then ℊ = −Re〈 ⃑ * • ℰ ⃑ 〉, and 〈•〉 indicates time average. Note that ⃑ = ⃑ + Re ⃡( ) • ℰ ⃑ , where ⃑ is spontaneous electric polarization, is vacuum permittivity, and ⃡( ) is optical susceptibility tensor at the frequency , containing electronic and phononic contributions ( ⃡ = ⃡ + ⃡ ).
Under LPTL illumination (along the i-direction), note that the light frequency is on the THz order, which could be lower than the phonon Debye frequency of the system.
Hence, if is large enough (much stronger than the coercive field to reverse polarization), the spontaneous polarization may follow the electric field and switch back and forth between ⃑ and − ⃑ . When the is much stronger than (corresponds to the situation in our current discussion), we can show that it contributes to the time-averaged GFE density in the form of (see Supplementary Note 3 for details) where ∈ [0, ] is the angle between the LPTL polarization direction and the spontaneous polarization direction. This is a first order interaction that measures between light and polarization. If the light frequency is highly above the phonon frequency range (e.g. several tens THz and above), then the ion position change cannot follow the light field (off-resonant). Then this term would become zero as the ⃑ is time-independent. We also include the second order interactions by incorporating optical susceptibility. This can be considered by the averaged GFE density change through integrating time and electric field in ℊ = − ℰ ⃑ * ( ) • Re ⃡ ( ) + ⃡ ( ) • ℰ ⃑ ( ), and the total GFE density contributed from optical linear responses can be written as Note that the integration of ℊ over electric field space gives one factor, and the According to the linear response theory with random phase approximation (RPA), the electronic part of susceptibility ⃡ ( = , ) is the Fourier transformation of real space density response function ⃡ ( , , ), which is solved from a Dyson equation where ⃡ ,( ) is bare density response function (independent particle) contributed by electron transitions Here, fi, ϵi, φi are the Fermi-Dirac distribution, energy, and wavefunction of the i-th level, and ξ is the Lorentzian phenomenological damping parameter (taken to be 0.025 eV in our calculations), representing disorder, finite temperature, and impurity effects.
When the frequency of LPTL lies in the range of phonon frequency (a few THz), vibrational phononic contribution to optical susceptibility needs to be included.
According to lattice dynamics theory 22 , the LPTL is coupled with infrared-active transverse optical (TO) mode of phonons at the Γ-point of the Brillouin zone. The phononic contributions to susceptibility is calculated according to 23 where Ω is the unit cell volume, α and β (=  24,25 .

Results.
Monolayer β-GeSe. We now apply these analyses to two experimentally fabricated 2D multiferroic materials. Figure 2(a) shows the atomic structure of β-GeSe monolayer, with relaxed lattice constants to be a = 3.59 Å and b = 5.73 Å. This structure belongs to Pmn21 space group (no. 31) without centrosymmetry, consistent with previous works 26,27 . One clearly observes that it looks like a distorted honeycomb lattice (such as h-BN, compressed along the y-direction). Actually, the honeycomb structure is serving as a parental phase (Supplementary Note 4), and the β-GeSe shown in Fig. 2  (b) Calculated real part of optical susceptibility and absorbance function with respect to incident energy. The gray shaded region indicates phononic range (< 8 THz), above which the responses are mainly contributed from electronic subsystem. The absorbance in the phononic range is enlarged twenty times for clarity reasons. The subscripts "1" and "2" denote the Cartesian coordinates "x" and "y", respectively.
In addition to ferroelastic order, the lack of centrosymmetry indicates a spontaneous electric polarization ⃑ . Consistent with previous works 27  Here, superscripts "3D" and "2D" refer to susceptibility in the supercell and in the 2D form, ∥ indicates that only xy-plane component can be scaled, and and ℎ are simulation supercell and 2D material thickness, respectively. We use the separation distance in the bulk structure to estimate ℎ, which gives 5.5 Å. The calculated ⃡ ∥ are shown in Fig. 2b. Detailed electronic and phononic properties can be seen in Supplementary Note 1. We find that above the phonon dispersion region (~8 THz, corresponds to 0.033 eV) and below the direct bandgap (1.6 eV), the = 25.2 > = 7.8 . This can be understood from anisotropic electronic transition strength, reflected by the imaginary part of susceptibility at the direct bandgap, which determines the absorbance. The absorbance is calculated by where is the speed of light and is imaginary part of susceptibility. The absorbance for the x-polarized and y-polarized light at 1.6 eV are 0.63% and 6.7%, respectively, owing to the saddle-like exciton feature and large anisotropic joint density of states. This is consistent with previous works 26 , and quantitative differences come from different formulae.
In the phonon frequency region, the TO phonon modes interact with LPTL strongly. The optical branches of vibrational modes at the Γ-point can be decomposed according to the irreducible representations as We Here, ∥ and are wavevectors parallel and perpendicular to the electron movement direction ( ∥ = / ), respectively, = ∥ + , and is angle-dependent polarizability. When the system is isotropic, the polarizability reduces to its well-known form = . We plot the EELS spectra when the electron is moving along the x-and y-direction (Fig. 3). One could clearly observe large direction dependent EELS feature, especially in the phonon region. This anisotropic vibrational EELS originates from different infrared-active phonon characters of the x-and y-LPTL in the β-GeSe monolayer. This result provides a high resolution damage-free approach to distinguish the geometric structure and anisotropy when the ferroic order switches. Monolayer α-SnTe. The monolayer β-GeSe has three orientation variants with 120° rotation to each other, owing to the character of the parental geometry. Thus, even though the LPTL response is largely anisotropic (~66 times difference in the xand y-directions at incident frequency of 1 THz), the energies separating different orientation variants are on the order of 1 μJ/cm 2 . Now we consider another 2D time-reversal invariant multiferroic material, monolayer α-SnTe, whose parental geometry is C4 symmetric 39,40 . As shown in Fig. 4(a), the α-SnTe also shows a puckered structure, with slight expansion (compression) along the y-(x-)direction. Note that even though bulk and multilayered α-phase other group IV-VI compounds (such as α-GeS, α-GeSe, α-SnS, and α-SnSe) have been experimentally seen, their monolayer remains to be fabricated. Hence, we only focus on the α-SnTe monolayer, and similar results for analogues can be obtained. Our relaxation reveals that the structure also belongs to be the Pmn21 space group, and the deformation strain tensor of this ferroelastic structure  We employ DFT and DFPT methods to calculate the electron and phonon dispersions (Supplementary Note 1), and compute the optical susceptibility (Fig. 4b).
We find the electronic contributed optical response anisotropy in α-SnTe monolayer is not as large as that in the β-GeSe. At the direct optical bandgap (0.9 eV), the absorbances of the x-polarized and y-polarized light are 0.9% and 1.6%, respectively. Such barrierless phase transition does not require latent heat and can occur anywhere LPTL is shined. This indicates a spinodal-decomposition in the reaction coordinate space, avoiding the conventional nucleation-and-growth kinetics. Such process only requires one or a few vibrational oscillatory process, which is ultrafast and could occur on the order of several picoseconds 42 .
Note that this system is interesting as its spontaneous polarization favors to align perpendicular to the optical polarization direction. This is even correct when the LPTL frequency reduces to zero -a static electric field. Since under x-directional electric field (magnitude larger than 0.3 V/nm), the spontaneous polarization prefers to align along y, counterintuitive with the conventional E//P picture.
The direction dependent EELS spectra is also evaluated (Fig. 4c). We again observe that the vibrational EELS shows a large anisotropy. The main EELS spectra respectively. Specially, they can be evaluated by where the position matrix element is defined as  SnSe. It is well-known that the standard DFT calculations underestimates the electronic bandgap, hence in principles one has to adopt more accurate approaches, such as hybrid functional or many body calculations 48 . Unfortunately, this is extremely computational challenging because of nonlocal interactions included and very dense k-mesh needed.
In the previous calculations 47 , Wang and Qian adopted a scissor operator scheme to shift the DFT calculated bandgap to be consistent with the optical bandgap calculated through many body theory with exciton interaction correction. In our current study, we only focus on the relative strength of SHG responses and its anisotropy, and the scissor correction is not applied here. Therefore, in experiments the peaks of the SHG responses would shift to higher energy compared with our calculation results, usually by ~ 0.5 -1 eV. According to Eqs. (10)-(12), the SHG peak magnitude roughly scales as 1/Eg (Eg is the electronic bandgap). Hence, the experimentally measured SHG peak values would also be smaller than our calculation results. Nevertheless, the overall SHG shape and anisotropy trend should be similar with our calculations.
Discussion. In our current LPTL driven phase transition mechanism, we only elaborate the real part of optical susceptibility and assumes its imaginary part to be small. Actually when the optical frequency is chosen around finite Im(χii), direct optical absorption occurs. This corresponds to a conversion from photons to infra-red phonon or electron-hole pairs (electron interband excitation). Actually this could also trigger mechanical strains 49 to the systems, and phase transition may occur 50,51 . However, they would generate unwanted heat into the system via phonon-phonon interaction or nonradiative electron-hole recombination so that the device reversibility will reduce. In our mechanism, only electric field work done is applied to the systems. Even though all of the work done converts to be internal energy, the temperature rise is still very small 30 .
Hence, in the current discussion, we only focus on the real part of optical susceptibility contribution and avoid such direct photon absorption process.
In addition, we note that in the current setup only ferroelastic order degeneracy can be broken under LPTL irradiation. One could not break the degeneracy between the ferroelectric phases with opposite polarization states ( ⃑ and − ⃑ ). Thus, one needs to apply additional fields (such as zero frequency static electric field or introducing surface/interface effects) to break such degeneracy. Actually, optical control of the However, an exact and complete evaluation is very complicated, and is out of the scope of our current study.
The phase transition related strain in the current study is within 8%, which is usually sustainable for 2D materials. However, one may notice that if the sample size is a few to a few tens of nanometers, then such strain would cause a big lattice mismatch in the system, and may even affect the chemical bonds at the boundary between the transformed and untransformed domains. A direct numerical simulation of such process in a large area is very computationally challenging and memory demanding. Actually one may allow a freestanding 2D materials to be slightly slack in the z-direction, or carefully select a surface to support them with weak interactions (such as van der Waals) 54 . This allows the atoms to move in the z-direction with small energy cost, which could effectively release the in-plane strains during martensitic phase transitions. This is different from 3D bulk materials or thick films, where such strain induced damage can be significantly large and is fatal to their reversible usage.

Conclusion.
In conclusion, we implement a theoretically and computationally combined approach to study the electronic, phononic, and mechanical responses of 2D According to the thermodynamic theory, we predict that LPTL can efficiently drive phase transition with an ultrafast kinetics (or even a barrierless GFE profile). This noncontacting optomechanical approach to switch the ferroelastic order of 2D materials can be easily controlled experimentally. In order to detect different orders, we propose to measure vibrational EELS spectra, which is direction dependent and has an ultrahigh resolution experimentally. Anisotropic SHG response calculations are also provided.
Different from the parameterized phonon-phonon coupling models in the optically induced phase transition, we provide a first-principles quantitative estimation on the terahertz optics effects. Such mechanism can also apply to other frequency range, as long as the direct optical absorption is eliminated. Owing to the rapid developments of using terahertz laser triggering (topological or structural) phase transition, our theory provides a route to explain these experiments and predict unexplored phenomena from a precise first-principles approach.
Methods. Our first-principles calculations are based on density functional theory 14 and performed in the Vienna Ab initio Simulation Package (VASP) 15 with generalized gradient approximation (GGA) treatment of exchange and correlation functional in the solid state PBE form (PBEsol) 16 . In order to simulate two-dimensional (2D) materials, a vacuum distance of 15 Å in the out-of-plane z-direction is applied to eliminate layer interactions. The projector augmented wave method 17

I. Supplementary Note 1: Electronic and phonon dispersions and contributed dielectric functions.
We plot the electronic band structure and phonon dispersion along the high symmetric k-path, of both FE0-β-GeSe and FE1-α-SnTe monolayer (Supplementary Fig.   1). One can see that the β-GeSe monolayer is an indirect bandgap semiconductor, with From phonon dispersions, we observe no imaginary frequencies over the whole Brillouin zone (BZ). This suggests their dynamic stability, and confirms that they can exist in free-standing monolayer forms. In addition, we plot the mass averaged vibrational modes at the Γ-point that contributed to the phononic susceptibility ( Supplementary Figs. 2 and 3). One sees the infra-red active feature of these modes.

II. Supplementary Note 2: Calculation details of spontaneous polarization
According to the modern theory of polarization, the exact electric polarization value is not well-defined, and one could only evaluate the polarization change according to the Berry phase approach, which measures the spontaneous polarization from a centrosymmetric structure Here we use the 2D Brillouin zone, and the scripts 'i' and 'f' refers to the initial and indicates the energy barrier (per f.u.) that separates the polarization reversal (between and − ) and . . is the area of one formula unit. One has to note that this energy barrier can be different from the energy difference between the high symmetry structure and ferroelectric ground state, and such ideal coercive field is very roughly estimated.
Actually one can also estimate the atomic position change according to a simple damped driven oscillator model. Under a sinusoidal electric field, the system oscillates according to where , , and * are average values of displacement, mass, and Born effective charge of ion-μ, respectively. is phonon frequency and is effective damping.
The displacement amplitude then can be evaluated by Since the phonon frequency is only 1.2 THz for GeSe and 1.4 THz for SnTe, we estimate that under electric field amplitude of = 0.2 V ⋅ nm at frequency 1 THz, the ion amplitude is on the order of 1 Å, which is sufficiently large for and − reversal. This also indicates that is much larger than , consistent with previous analysis.
In order to estimate the polarization reversal contributed Gibbs free energy (GFE), we plot a simple polarization reversal process as in Supplementary Fig. 4. One could consider the following two steps (for simplicity, we first take 1D model): (i) The sinusoidal electric field ℰ( ) = sin is between − to − and to (From step  to  to ): Once an electric field − is applied, the system jumps to polarization − . It contributes to the GFE to be − . . ℰ( ). Note that the linear response (which depends on susceptibility ) is not included here, which will be discussed in the main text.
(ii) The electric field between + to + and back to − ( to  to ).
This process is a reversal process as (i), and gives − . . ℰ( ) to GFE.
Supplementary Figure 4. THz optics driven polarization reversal and its contribution to Gibbs free energy. Left panel is a simplified E-P plot of ferroelectrics, and the sinusoidal electric field variation is plotted in the right panel. The slope of the green line in the E-P plot corresponds to the optical susceptibility that contributed from both electron and phonon subsystems. The two horizontal dashed lines indicates the critical electric field ±Ec.
One could then integrate over time of the whole process to obtain the total GFE contribution. It is easy to see that between the two dashed horizontal lines in the right panel of Supplementary Fig. 4, the contribution to GFE cancels each other. Thus, only the → → and − → − → − processes give finite GFE, which are the same to each other. One easily obtains the results to be In our current case, the critical electric field (on the order of 0.01 V⋅nm −1 ) is order of magnitude smaller than the optical field strength, thus the above equation can be approximated to be ℊ = − . If the electric field is not parallel to the polarization direction, one can decompose the electric field into ∥ = cos and = sin , where ∈ [0, ] is the angle between the LPTL polarization and the polarization .
The above equation then becomes This is the Eq. (1) in the main text. Note that if the light frequency is highly above the intrinsic vibration frequency, or the light electric field strength is smaller than the coercive field, such ℊ term becomes zero, and the only optical susceptibility term contributes to the total GFE density.

IV. Supplementary Note 4: Ferroelastic phases of β-GeSe monolayer.
We use DFT calculations to compute the different ferroelastic orders of β-GeSe monolayer ( Supplementary Fig. 5). In order to evaluate the spontaneous transformation strains of each ferroelastic order, we use a (2 × 2√3) supercell of the high symmetric parental structure, and all the three ferroelastic orientation variants can be derived by We apply Eqs. (1) and (2) in the main text to compute the Gibbs free energy under LPTL illumination with different polarization angle. In Supplementary Fig. 6, we plot the polarization angle dependent Gibbs free energy change of FE0-β-GeSe monolayer.
The angle is defined as between the polarization direction and the armchair direction (y in FE0-β-GeSe). One could observe that, due to small x-component phononic susceptibility, the Gibbs free energy reduction is higher when an x-polarized terahertz laser is applied ( = 90°). If the initial orientation variant is FE0, one could apply a LPTL with its polarization along (with 〈 , 〉 = ± ) to switch it to FE1 or FE2.
Supplementary Figure 6. Angle-dependent GFE change of GeSe monolayer. The laser frequency is selected to be 1 THz, and the magnitude of electric field is 0.2 V⋅nm −1 .
The angle is defined as between the LPTL polarization direction and the GeSe armchair direction.

VI. Supplementary Note 6: Electron energy loss spectroscopy (EELS).
EELS can provide geometry information, which uses an electron beam positioned a few tens of nanometers away from the sample ("aloof" mode). Compared with the standard transmission experiment with the same electron energy, this noninvasive and aloof mode reduces the sample damage by a factor of ~1000 (Supplementary Ref. 3).
We can combine electromagnetic dynamics theory within a continuum dielectric model to obtain the EELS spectrum intensity as a function of electron energy loss for 2D materials.

Supplementary Figure 7.
Geometric model for fast electron travelling beside a 2D material. The normal direction to plane is denoted as z. Trajectory is marked by the blue arrow.
Supplementary Fig. 7 shows a simplified geometric model when a fast electron travels parallel to a 2D material surface (sandwiched by vacuum in both sides).
According to non-relativistic electrodynamics, the electric potential generated by this electron can be written as (SI unit) The = ∥ + denotes the surface excitation wavevector. From this, we can write the potential in the three regions.
where E is the electric field induced by the polarizability of 2D material, and re is the electron position. From the above equation, the energy loss probability can be calculated as The EELS is measured as energy-loss probability per unit angular frequency and per unit path length   respectively. Under LPTL, the FE2 has lower Gibbs free energy than the FE1, indicating a FE1 to FE2 phase transition. In particular, when the laser intensity is increased to 2.2×10 10 W⋅cm −2 (corresponding to electric field magnitude of 0.42 V⋅nm −1 ), we find that ≤ < . Thus, it is a barrier-free phase transition kinetics.
As stated in the main text, such barrier-free phase transition could occur ultrafast and does not require latent heat. The process can avoid the conventional nucleation-andgrowth kinetics, which is important for fast memory read/write technology.

VIII. Supplementary Note 8: Direction dependent SHG susceptibility.
The space group of both GeSe and SnTe has ℳ mirror symmetry. Thus, if the light polarization direction is along = (cos , sin ) , then the SHG susceptibility parallel (∥) and perpendicular (⊥) to can be written as (note that = ) ∥ ( ) = + sin cos 2 +

IX. Supplementary Note 9: Microscopic mechanism of optics phonon interaction.
In the main text, the interaction between light and matter is discussed according to macroscopic thermodynamic approach, which could directly and straightforwardly provide the kinetics of phase transition. In this section, we briefly discuss its microscopic mechanism according to quantum mechanical perturbation theory 6 . The electric field can be written as The semi-classical expression of induced electric moment between states | ⟩ and | ⟩ can be calculated as The interaction between electric field and moment serves as an additional perturbation term in the Hamiltonian The wavefunction Ψ can be expanded according to perturbation theory, where ℏ and are the l-th eigenvalue and wavefunction of unperturbed system, respectively. | ⟩ is a virtual state, = 1,2,3 represents three Cartesian directions.
The electric moment then takes the form The Π ( ) is transition polarizability, which indicates a non-coherent transition from | ⟩ and | ⟩ Π ( ) = ℏ ∑ + This process is illustrated in Supplementary Fig. 10.
Supplementary Figure 10. Brief illustration of photon induced deformation. The solid lines represent real state that system could stay for sufficiently long time, unless there is dissipation. The dash-dotted line represents a virtual state. Classically, it corresponds to an oscillating ion or electron subsystems.
According to Placzek adiabatic approximation, the total wavefunction Ψ can be divided into ion wavefuntion and electron wavefunction , Here and are electronic and ionic quantum numbers, and represent electron and ion positions, respectively. Thus, the | ⟩ and | ⟩ states can be written as | ⟩ and | ′ ′⟩, respectively. The electronic ground state is = 0. The transition polarizability can be written as The contribution can be divided into two parts, namely, electronic and ionic contributed transition polarizability Π ( ) = Π ,( ) ( ) + Π ,( For the ionic contributed part, the electron always stays at its ground state when the ion transits from | ⟩ to virtual | ′′⟩ and drops back to | ′⟩. Thus,