Quantum Geometric Injection and Shift Optical Forces Drive Coherent Phonons

The gist The injection and shift rectified Raman forces, which are phononic counterparts of the photogalvanic effect, drive lattice vibrations and trigger transient emergent properties.

Quantum Geometric Framework

The study introduces a quantum geometric framework for unconventional light-induced atomic forces by introducing the terms injection force and shift force by analogy with established second-order photocurrent processes The injection force is described as the sole displacive component and arises only in the absorbing (resonant) regime, where it is directly described by the QGT The shift force acts as an unconventional resonant impulsive force in the absorbing regime when time-reversal symmetry is broken The QGT can be decomposed into its real and imaginary parts as Qmn = gmn − i 2omegamn, where the real part, gmn, represents the (band-resolved) quantum metric (QM), and the imaginary part, omegamn, represents the (band-resolved) Berry curvature (BC).

Force Components and Mechanisms

The paper details two distinct forces:

1. Injection force: This is described by Finj ∼ gvc(Mv − Mc) It is the sole displacive component and arises only in the absorbing (resonant) regime, where it is directly described by the QGT.

2. Shift force: This force emerges impulsively in the resonant interband absorbing regime when time-reversal symmetry is broken. It is formulated in terms of the QGT and a phononic shift vector, which measures the resulting change in ionic momentum from electron transitions between bands.

Response Functions and Symmetry

The rectified Raman force is given as a second-order response to the incident light field E(t): Fa(Ép) = Z dÉÇabc(É, Ép − É)E b (É)E c (Ép − É), (1) The analytical properties of the second-order response function Çabc has four contributions: two resonant terms from the Feynman diagrams of Fig. 2 and two off-resonant terms. For linear polarized (LP) light, the injection and shift forces are given in terms of the QM tensor by Eq. (4). For circular polarized (CP) light, the imaginary part of the QGT, that is the BC (given by omeganm bc = − 1/2 Im[Anm b Amn c]), will also emerge in the response and the shift force will depend on a chiral phononic shift vector. The RF exhibits distinct symmetry characteristics relative to the second-order photocurrent.

Bilayer Haldane Model Results

The injection and shift forces were quantitatively computed in the bilayer Haldane model, which breaks T-symmetry via magnetic flux and P-symmetry via sublattice staggering. The numerical results for the injection and shift shear RF responses to LP light are depicted in Fig. 3. The injection force is largest in the gapless regime and diminishes as the Haldane gap increases. On the other hand, panel (b) displays the shift response, which shows an odd-parity dependence on the Haldane gap.

Off-Resonant Contributions

The off-resonant force analysis reveals that both parts of the off-resonant Raman force are not treated in the main text. The first part, arising from the principal value terms in the clean limit, yields 3off abc(É) = 0, indicating that there is an absence of an off-resonant counterpart to the injection force. The second part, originating from the oval diagrams, is written in terms of the quantum geometric tensor Qnm bc.

Phononic Shift Vector Calculation

The phononic shift vector was obtained by finding an alternative form for Eq. (S38). The expression for the chiral phononic shift vector is given by Rmn a;± = − Im(Mmn a;v nm ∓) + ImP l Mml a v ln b v nm b εml− Mln a vml b v nm b εln− vmn b 2, (S61). This vector can be easily calculated numerically.

CP Response and Symmetry Breaking

For CP light, the shift force susceptibility becomes ´shi−CP abc(É) = Ãe2 4ħ X p X mn fmn (g mn bb + g mn cc)(Rmn a;+ − Rmn a;−) − omega mn bc (Rmn a;+ + Rmn a;−) ¶(É − Énm), (S45). The presence of C3 symmetry forbids these CP resonant responses in the bilayer Haldane model, necessitating an additional perturbation like a unidirectional source-drain DC current flow to break this symmetry. The finite current flow then induces nonvanishing susceptibilities, which increase with the magnitude of the current.

Numerical Results for CP Forces

The numerical results for the CP injection and shift shear force susceptibilities are shown in Fig. S3. Three distinct features are seen in the spectra for both CP injection and shift shear forces in Fig. S3, which are, from left to right: a peak at ħÉ = 2µ corresponding to the UVB-to-LCB transition, a peak derivative-like feature associated with the LCB-to-UCB transition, and a shallow peak from the LVB-to-LCB transition. This current-induced shear force is driven by the current-induced stationary population distortion, ¶fm, which is significant only near the Fermi surface. The fact that all three features exhibit sharp, peak-like behavior clearly originates from the fact that all listed transitions involve the LCB.

Cancellation of Counter Terms

In the intrinsic geometric limit, we obtain an identically exact cancellation as follows µ △ abc(É) + µ O abc(É) = 0, (S49). This reasoning can be compared to the cancellation of the counter terms for the injection and shift photocurrents. The clean limit protocol captures the cleansystem cancellation of the first and second lines in the above expression, independent of the magnitude of the band gap.

Off-Resonant Force Analysis

The off-resonant contribution from diagrams (c) and (d) is given by ´off,2 abc(É) = −e squared 2É squared X p X m fmMmm a v nm bc. This expression shows that the second term includes the phononic dipole of quantum metric similar to that in Eq. (S52). The first term resembles a higher-order Drude response, which describes the Drude DC photocurrent if the electron-phonon coupling operator is replaced by the velocity operator.

Numerical Accessibility of Shift Vector

The numerical calculation of the shift vector can be performed using Eq. (S62). This leads to Rmn a;± = − Im(Mmn a;v nm ∓) + ImP l Mml a v ln b v nm b εml− Mln a vml b v nm b εln− vmn b 2, (S62). This expression is easily calculable numerically.

1. Improved AI systems can perform ultrafast, coherent manipulation of quantum materials by controlling lattice vibrations using light-induced forces. Specifically, these systems can drive ultrafast phase transitions to transient phases such as from a normal metal to a topological insulator, or control interlayer shear phonons in layered materials like bilayer graphene.

2. AI systems can precisely tune the magnitude and direction of rectified Raman forces by varying external parameters like driving frequency and magnetic flux. This allows for the discovery of a distinct quantum geometric mechanism for ultrafast and coherent manipulation of quantum materials.

3. Improved AI systems can analyze the topological characteristics of electronic states, specifically utilizing the band-resolved quantum geometric tensor (QGT), to predict how these features affect light-induced atomic forces. This enables understanding how the nature and sign of different components of the RF can be affected by the topological characteristics.

4. AI systems can model and simulate complex quantum mechanical phenomena using frameworks like the Kadanoff-Baym formalism to derive susceptibilities for injection and shift forces under various light polarizations (LP and CP). This allows for predicting the resonant component of these forces, which is crucial for coherent phonon dynamics.