Instantaneous Optical Valley Selection Rule

• Instantaneous optical valley selection rule is a framework where allowed electronic transitions are determined by the instantaneous polarization or chirality of light, enabling time-resolved valley control.
• It underpins phenomena in monolayer TMDs and Dirac materials by linking circular polarization to valley-specific excitation and incorporating quantum geometric effects.
• New formulations allow sub-cycle chirality control and selective access to unconventional valleys, promising near-100% valley-polarized currents for advanced valleytronics.

Instantaneous optical valley selection rule denotes a class of optical selection rules in which the allowed transition at a specific valley is determined by the polarization, helicity, or chirality of light at the moment of absorption or emission. In its conventional form, it is the valley-contrasting mapping between circular polarization and inequivalent valleys such as $K$ and $K'$; in newer formulations it also includes sub-cycle selection by instantaneous optical chirality, orthogonal linear-polarization selection governed by the quantum metric, excitonic and moiré channels with opposite helicities, and proposed access to a single $Q$ valley in monolayer transition metal dichalcogenides (TMDs) (Jones et al., 2013, Li et al., 12 Jul 2025, He et al., 11 Aug 2025, Kim, 10 Aug 2025).

1. Conventional valley–helicity locking

The canonical realization appears in monolayer TMDs, where bright excitons in opposite valleys couple to opposite photon helicities. In monolayer WSe$_2$, the optically active mapping is stated as

$$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$

and linearly polarized excitation acts as a coherent superposition of $\sigma^+$ and $\sigma^-$, thereby generating valley coherence rather than only valley polarization. The experimentally observed linear photoluminescence of the neutral exciton, with polarization axis tracking the excitation axis, is interpreted as recombination from a coherent superposition of the two valleys rather than from a single-valley population (Jones et al., 2013).

In spin-orbit-coupled Dirac materials, the same principle acquires a spin-valley form. In silicene, right- and left-circularly polarized light excite different valley-spin sectors, and the rule changes across the topological insulator to band insulator transition because band inversion reverses the band ordering near the gap. At $k=0$, the valley-resolved optical matrix elements satisfy

$$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$

and the circular dichroism changes sign at the phase transition (Ezawa, 2012).

The same dependence on broken symmetry and topology appears in a minimal spin-orbit-coupled honeycomb model. There, circularly polarized light selectively excites specific spin-valley sectors, and the active sector switches abruptly at the topological transition $h_c=\lambda/2$. In the charge-ordered and $K'$0-axis spin-ordered states, the optical rule is therefore not fixed solely by crystal symmetry; it is also contingent on the topological character of the insulating phase (Yanagi et al., 2017).

In the quantum Hall regime, monolayer MoS$K'$1 retains valley-contrasting optical selection even after Landau quantization. The inter-Landau-level rules become

$K'$2

$K'$3

so the photon helicity remains locked to the valley index rather than being washed out by the generic $K'$4 Landau-level rule familiar from graphene (Chu et al., 2014).

2. Symmetry, orbital texture, and quantum geometry

Microscopically, instantaneous valley selection originates from broken inversion symmetry, time-reversal constraints, and the orbital angular structure of Bloch states. In the direct-gap monolayer MA$K'$5Z$K'$6 family, the band-edge basis is written as

$K'$7

with $K'$8 for $K'$9. The circular optical matrix element

$Q$0

then yields valley-dependent interband transitions under circularly polarized infrared light (Yuan et al., 2021).

In nonlinear optics, the valley contribution can be formulated as a distinct tensor component. For monolayer WSe$Q$1, the second-order valley-polarization susceptibility is written as

$Q$2

so the nonlinear response changes sign with valley helicity. Under circular excitation, the second-harmonic polarization contains both the intrinsic electric-dipole term and the valley term, and the interference between them provides a probe of broken time-reversal symmetry and of the relative phase between $Q$3 and $Q$4 (Herrmann et al., 2023).

A more recent generalization replaces Berry-curvature-only reasoning by a quantum-geometric formulation for linearly polarized light. The degree of linear polarization at a valley is

$Q$5

with $Q$6 corresponding to complete selectivity. When two valleys are related by a mirror or mirror-like symmetry, the theory gives

$Q$7

so orthogonal linear polarizations are locked to distinct valleys. This directly challenges the common restriction of valley selection rules to circular dichroism and Berry curvature alone (Li et al., 12 Jul 2025).

3. Sub-cycle chirality and optical-cycle control

A strong-field formulation makes the adjective “instantaneous” literal. In the sub-cycle theory of valley current control, the interband excitation phase contains the term

$Q$8

where $Q$9 is the instantaneous polarization direction. The optical field chirality is defined by

$_2$0

while the material chirality is encoded by $_2$1, with

$_2$2

The valley population asymmetry for electrons ionized at time $_2$3 is then

$_2$4

The sign of $_2$5 determines which valley is favored at that instant (He et al., 11 Aug 2025).

This formulation shifts valley optics from cycle-averaged helicity to the instantaneous rotation direction of the driving field. A chirality-separated optical field, synthesized from two co-rotating bicircular fields of opposite helicity, separates positive and negative instantaneous optical chiralities within one optical cycle. The resulting protocol enables independent manipulation of currents from $_2$6 and $_2$7 valleys, including complete separation of currents from different valleys yielding 100\%-purity valley-polarized currents, and generation of pure valley current with zero net charge flow (He et al., 11 Aug 2025).

The physical distinction from conventional valley pumping is that the two valleys are not driven by the same sub-cycle lobe. One lobe can populate $_2$8, another $_2$9, and the associated vector-potential asymmetry can steer the two valley currents independently within the optical-cycle timescale. This suggests a genuinely time-local form of valley selection rather than a time-integrated helicity bias.

4. Access to unconventional valleys and finite-momentum excitons

The extension from $$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$0 to other valleys is a major recent development. “Light-Wave Engineering for Selective Polarization of a Single $$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$1 Valley in Transition Metal Dichalcogenides” states that coherent combination of a circularly polarized pump pulse with a linearly polarized driver pulse yields an emergent light-wave valley selection rule that enables deterministic and high fidelity excitation of any single $$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$2 valley in monolayer TMDs, completely decoupled from the conventional $$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$3 valleys. The abstract further claims near-unity ($$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$4) valley polarization across an exceptionally broad ultrafast window, from the terahertz ($$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$5) to petahertz ($$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$6) regimes, enabling single $$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$7 valley polarization on femtosecond timescales. The microscopic model is a three-band tight-binding Hamiltonian in the $$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$8, $$|K\rangle \leftrightarrow \sigma^+,\qquad |{-K}\rangle \leftrightarrow \sigma^-,$$9, and $\sigma^+$0 basis, with density-matrix dynamics

$\sigma^+$1

and

$\sigma^+$2

with $\sigma^+$3 for MoS$\sigma^+$4 (Kim, 10 Aug 2025).

Instantaneous selection rules also arise for finite-momentum excitons. In twisted TMD heterobilayers, interlayer excitons are optically active only at six finite-velocity light cones located at

$\sigma^+$5

At each light cone, the emitted photon is elliptically polarized, the major axis is locked to the direction of exciton velocity, and the helicity is specified by the valley indices of the electron and the hole. The selection rule is therefore attached to a finite center-of-mass momentum channel rather than to a zero-momentum direct transition (Yu et al., 2015).

A plausible implication is that “instantaneous” valley selection should not be restricted to zone-corner single-particle optics. It can instead be anchored to whichever momentum-space manifold—band extrema, finite-$\sigma^+$6 excitons, or higher-lying valleys—dominates the optically active transition.

5. Excitonic and nonlinear extensions

Excitonic structure introduces additional angular-momentum channels beyond the single-particle valley index. In monolayer WS$\sigma^+$7, the valley-exciton locking rule is expressed as

$\sigma^+$8

where the threefold lattice can absorb or supply angular momentum in multiples of $\sigma^+$9. This produces distinct two-photon rules for second-harmonic generation and two-photon luminescence: under $\sigma^-$0 excitation, the SHG helicity is approximately $\sigma^-$1 at the $\sigma^-$2 resonance, while the TPL helicity at the $\sigma^-$3 resonance is $\sigma^-$4. The measured dynamics give $\sigma^-$5, $\sigma^-$6, intervalley scattering during relaxation of $\sigma^-$7, and persistence of valley imbalance during recombination of $\sigma^-$8 (Xiao et al., 2015).

The excitonic Rydberg series in massive Dirac cones further modifies one-photon selection. In the simplest massive-Dirac description, $\sigma^-$9-like excitons are bright with one helicity in a given valley, while $k=0$0-states are bright with the opposite helicity in the same valley. Including trigonal warping brightens $k=0$1-states as well. The symmetry criterion is the discrete $k=0$2 rule

$k=0$3

which governs both one-photon generation from the vacuum and intra-excitonic transitions (Gong et al., 2017).

Nonlinear high-lying-band excitation provides yet another variant. In MoTe$k=0$4, a doubly resonant cascaded pathway from the valence band to high-lying states proceeds through a real intermediate state, so the effective nonlinear rule becomes the product of two one-photon rules, $k=0$5 and $k=0$6. In time- and angle-resolved extreme-ultraviolet photoemission, the valley asymmetry of the high-lying $k=0$7 feature is about 3 times larger than that of the first conduction band, and the time-dependent Lindblad formalism reproduces the short-lived character of this cascaded signal (Courtade et al., 5 May 2026).

6. Interpretation, diagnostics, and broader regimes

A recurrent interpretive issue is the difference between instantaneous selectivity and time-integrated observables. In electrically tunable moiré excitons of WSe$k=0$8/WS$k=0$9, a nearly zero steady-state valley polarization does not necessarily indicate fast valley relaxation. Helicity-resolved time-resolved photoluminescence reveals a temporal crossing between co- and cross-circular emission because A-like and B-like moiré channels emit opposite helicities and have different effective decay and depolarization rates. The two-channel description gives

$$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$0

and the crossing time

$$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$1

Time-integrated circular polarization can therefore be a false-negative indication of valley polarization in multichannel valley emitters (Urano et al., 26 Jun 2026).

Spatial separation offers a complementary diagnostic. In Bernal bilayer graphene, trigonal warping produces valley-dependent anisotropic photoexcitation, so carriers from $$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$2 and $$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$3 propagate to different sides of the light spot under linearly polarized excitation. In the gapped case, the recombination light from the two sides has opposite circular polarizations, which the work identifies as an optical valley Hall effect (Osborne et al., 2024).

Floquet engineering provides a different mechanism. In gapped graphene under high-frequency circularly polarized light, the renormalized gap is

$$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$4

The valley-odd term proportional to $$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$5 means that one valley experiences gap closing while the other experiences gap opening, so the illuminated region acts as a valley-selective trap or barrier, and reversing helicity swaps the selected valley (Dini et al., 2018).

Localized states supply an impurity analogue. For impurity-to-conduction-band absorption in gapped Dirac materials, the strongest valley selectivity occurs for orbital impurity states satisfying

$$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$6

especially $$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$7, while $$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$8 is much less selective because the threshold matrix element is suppressed. The selection rule comes from interference among impurity orbital phase, conduction-band chirality, and the helicity of circularly polarized light (Ko et al., 2020).

Taken together, these results establish instantaneous optical valley selection rule as an umbrella concept rather than a single formula. It includes band-edge helicity locking, topology-dependent spin-valley selection, quantum-metric linear dichroism, sub-cycle chirality control, excitonic angular-momentum rules, finite-$$P_{+}^{K}(k)=P_{-}^{K'}(k),\qquad P_{-}^{K}(k)=P_{+}^{K'}(k),$$9 light-cone selection, Floquet gap engineering, impurity-state selection, and proposed access to single-$h_c=\lambda/2$0 valleys. A plausible implication is that future valleytronic diagnostics will increasingly need to be time-resolved, momentum-resolved, and channel-resolved, because the instantaneous rule can remain sharp even when steady-state polarization is small or vanishing.