Quantum-Geometric Shift Current

• Quantum-Geometric Shift Current is a gauge-invariant rectified dc response arising from resonant interband transitions encoded by Bloch state geometry and shift vectors.
• A projector calculus framework combined with Berry connections and quantum geometric tensors unifies its description across multiband, excitonic, thermal, and finite-momentum regimes.
• Advanced computational models and experimental studies, especially in twisted bilayer graphene, reveal enhanced photovoltaic effects and novel rectification mechanisms.

Quantum-geometric shift current denotes a class of rectified dc responses in which a single quantum event produces a gauge-invariant displacement governed by the geometry of Bloch states. In its canonical form, it is the shift-current contribution to the bulk photovoltaic effect: linearly polarized light excites an interband transition, and the electronic wavepacket undergoes a real-space displacement encoded by the shift vector, a combination of Berry connections and the momentum derivative of an interband transition phase. Recent work has recast this response in explicitly gauge-invariant projector language, linked it to polarization cumulants and Wilson loops, and extended the same geometric logic to multiband, excitonic, photon-drag, thermal, and static-bias settings (Resta, 2024, Avdoshkin et al., 2024, Kitamura et al., 2 Jul 2026).

1. Canonical optical response and the shift vector

In a noncentrosymmetric crystal, the optical shift current is a second-order dc photocurrent,

$$J_a(0)=\sigma_{abc}(0;\omega,-\omega)E_b(\omega)E_c(-\omega),$$

with the standard interband form

$$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$

A common convention writes the shift vector as

$$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$

with intraband Berry connection

$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$

and interband dipole matrix element

$$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$

(Resta, 2024).

This formulation makes the mechanism explicit: the current is not a drift current driven by a dc bias, but a rectified polarization current tied to the real-space displacement of charge during a resonant interband transition. The shift vector is gauge invariant because the phase gradient of the interband dipole compensates the gauge dependence of the Berry connections. In this conventional $q=0$ optical setting, inversion symmetry must be broken, whereas time-reversal symmetry need not be broken; the response is distinct from the injection current, which is associated with asymmetric carrier injection and carries a relaxation-time dependence in the standard formulation (Ezawa, 1 Jul 2025).

Equivalent formulas can be written in length gauge or velocity gauge, with generalized derivatives of Berry connections or velocity matrix elements. In localized Gaussian-basis implementations, the conductivity is expressed through gauge-covariant generalized derivatives,

$$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$

and both length-gauge and velocity-gauge formulations were shown to agree closely when sufficiently complete basis sets are used (García-Blázquez et al., 2023).

2. Gauge-invariant geometry beyond the textbook shift vector

A major development is the replacement of gauge-dependent band-eigenvector expressions by projector calculus. Writing the Bloch Hamiltonian as $$H(\mathbf{k})=\sum_n E_n(\mathbf{k})P_n(\mathbf{k})$$, one can express the shift-current conductivity directly through the quantum Hermitian connection

$$C^{\mu;\alpha\beta}_{ab}=\mathrm{Tr}\!\left[P_b\,P_a^\alpha\left(P_b^{\mu\beta}+P_a^\mu P_b^\beta\right)\right],$$

leading to the compact gauge-invariant result

$$\sigma^{\mu\alpha\beta}_{\mathrm{sh}} = -\frac{i\pi e^3}{\hbar^2}\int [dk]\sum_{a,b} \left(C^{\mu;\alpha\beta}_{ab}-C^{\mu;\beta\alpha}_{ba}\right) f_{ab}\,\delta(\hbar\omega+\varepsilon_{ab}).$$

For time-reversal-symmetric systems under linearly polarized light, the observable conductivity reduces to the imaginary, symmetrized, subspace-antisymmetrized part of $$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$0 (Guo et al., 11 Sep 2025).

Within this projector framework, the shift current is no longer merely a two-band phase-derivative formula. It becomes part of a broader hierarchy of explicitly gauge-invariant tensors, including the quantum geometric tensor and the triple phase product,

$$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$1

which organizes multiband nonlinear optics. This formulation also clarifies degeneracies: one works with projectors onto degenerate subspaces, so no gauge fixing within a degenerate manifold is required (Guo et al., 11 Sep 2025).

The same projector calculus shows that the frequency-integrated linear shift current decomposes into two pieces: a term determined solely by the occupied projector and a genuinely multi-state term involving virtual transitions through unoccupied bands. For a single filled subspace $$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$2, the relevant band sum is

$$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$3

The first term is directly related to the third cumulant, or skewness, of the electronic polarization; the second is intrinsically multi-state geometry and vanishes only in strict two-band settings (Avdoshkin et al., 2024).

A complementary Wilson-loop viewpoint identifies the shift vector as a geodesic-curvature-like object in momentum space. For a closed loop on a smooth two-dimensional manifold $$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$4 in the Brillouin zone, the shift vector and Berry-curvature difference satisfy a Gauss-Bonnet-like relation,

$$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$5

with $$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$6 an integer-valued interband character. In this formulation, the shift vector is interpreted as the gauge-invariant rate of turning of an interband state trajectory, and its loop integral contributes the non-quantized part of the circular photogalvanic trace when no monopole is enclosed (Wang et al., 2024).

3. Beyond vertical optical transitions: scattering, finite momentum, and static-bias analogues

The same geometric structure reappears when the transition is not a vertical optical excitation. In the thermal phonogalvanic effect, a hot electron gas in a noncentrosymmetric quantum well relaxes by inelastic electron-phonon scattering, and each scattering event carries a coordinate shift

$$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$7

the direct scattering analogue of the optical shift vector. When the electron and phonon baths have different temperatures, emission and absorption do not balance, so the event-by-event displacements fail to cancel and a dc current linear in $$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$8 emerges. In zinc-blende quantum wells this current vanishes for [001] and [111] growth axes and, in the $$\sigma_{abc}(0;\omega,-\omega)=\frac{\pi e^3}{\hbar^2}\sum_{nm}\int_{\mathrm{BZ}}\frac{d^dk}{(2\pi)^d}\,[f_n(\mathbf{k})-f_m(\mathbf{k})]\,R_{nm}^a(\mathbf{k})\,r_{nm}^b(\mathbf{k})\,r_{mn}^c(\mathbf{k})\,\delta(\omega_{mn}(\mathbf{k})-\omega).$$9 geometry considered, scales as $$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$0 (Budkin et al., 2019).

At finite optical wavevector $$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$1, photon drag unlocks shift-current physics even in centrosymmetric materials. The photon-drag shift current is the $$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$2 part of the second-order conductivity and can be written in terms of the interband quantum-geometric tensor

$$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$3

For linearly polarized light it is governed by the interband quantum metric $$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$4 and the Berry-curvature dipole, while for circular polarization it is governed by the Berry curvature and the quantum-metric dipole. In nonmagnetic centrosymmetric crystals, symmetry isolates the photon-drag shift current in the $$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$5 polarization channel under oblique incidence, providing a route to observe shift-current physics without static inversion breaking at $$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$6 (Xie et al., 28 Feb 2025).

A related finite-$$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$7 regime appears in Dirac electrons. There, particle-hole symmetry suppresses the conventional Fermi-sea contribution, and the circular shift photon-drag current becomes a dissipationless Fermi-surface response controlled by the dipole of the quantum metric tensor. For the massive Dirac model one has

$$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$8

and the helicity-dependent photon-drag conductivity is proportional to this metric dipole. The response is enhanced by a small band gap and remains finite in the massless limit; an explicit application to the $$R_{nm}^a(\mathbf{k})=\partial_{k_a}\arg r_{nm}^b(\mathbf{k})-\big[A_n^a(\mathbf{k})-A_m^a(\mathbf{k})\big],$$9-point Dirac Hamiltonian of bismuth shows that trivial-topology bands can still support a circular photon-drag effect through quantum geometry (Qu, 25 Mar 2025).

The geometric logic extends even further to a static-bias, nonreciprocal dc analogue. In the “shifted quasiequilibrium” picture, a Bloch wavepacket under a uniform dc field samples a tilted local chemical potential across its finite metric-controlled spatial extent, producing the quantum correction

$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$0

Contracting this with the band velocity yields

$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$1

This is not an optical shift current, but an intraband dc analogue governed by the quantum-metric dipole. Unlike the conventional optical shift current, it requires both inversion and time-reversal breaking, vanishes without dissipation, and in the constant-$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$2 model is independent of $$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$3 (Kitamura et al., 2 Jul 2026).

4. Multiband, excitonic, and higher-order generalizations

Multiband structure can enhance the shift current far beyond the resonant two-band picture. A gauge-invariant decomposition of the shift-current integrand separates direct resonant terms from virtual multiband terms involving intermediate bands $$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$4,

$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$5

This mechanism is amplified by inter-orbital mixing, small intermediate denominators, and multiple flat or quasi-flat bands. In stacked Rice-Mele chains and alternating-angle twisted multilayer graphene, the enhancement tracks the increase of the Fubini-Study metric and of the Wannier spread; for twisted multilayer graphene it produces strong low-THz peaks and identifies optimal twist angles distinct from the usual magic angles (Chen et al., 2024).

Electron-hole binding qualitatively reshapes the geometry of optical transitions. For excitons, the many-body shift vector

$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$6

becomes an intrinsic property of the correlated excitation and, for bound excitons, is independent of the optical polarization up to exponentially small finite-size corrections. A central consequence is that vertical excitonic transitions in nonpolar noncentrosymmetric crystals have vanishing shift vector and hence vanishing excitonic shift photocurrent, in sharp contrast with the polarization-dependent single-particle shift vector above the gap (Yang et al., 9 Jul 2025).

The formalism also generalizes to higher orders. In two-band models the $$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$7-th-order shift current is expressed through higher covariant derivatives of interband dipoles, summarized by a higher-order quantum connection,

$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$8

with

$$A_n^a(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_a}u_{n\mathbf{k}}\rangle,$$9

In the $$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$0-wave-magnet model analyzed in detail, the conventional second-order injection and shift currents vanish, while odd-$$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$1 higher-order injection and shift currents become nonzero for an in-plane Néel vector (Ezawa, 1 Jul 2025).

A further unification emerges in approximately linear, low-frequency band structures. In that regime, the linear shift current and the circular injection current become controlled by the same interband Berry-curvature dipole, while the circular shift current and the linear injection current become controlled by the same interband quantum-metric dipole. The resulting relations,

$$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$2

show that apparently distinct nonlinear optical responses can become equivalent probes of a common interband quantum geometry (Yahyavi et al., 9 May 2026).

5. Materials, model systems, and computational frameworks

Twisted bilayer graphene furnishes a particularly striking flat-band realization. In that system, the shift current decomposes into a conventional real-space piece and a dominant momentum-space piece,

$$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$3

where $$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$4 is built from derivatives of interband dipoles and interband Berry curvature and represents a momentum-space shift beyond the usual real-space picture. For twisted bilayer graphene with a gap of several meV, this produces a giant bulk photovoltaic effect in the $$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$5–$$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$6 THz range; for unstrained samples the transverse conductivity $$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$7 reaches nearly $$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$8, while in strained samples it reaches $$r_{nm}^b(\mathbf{k})=i\langle u_{n\mathbf{k}}|\partial_{k_b}u_{m\mathbf{k}}\rangle$$9 at $q=0$0 K (Kaplan et al., 2021).

At the device level, time-dependent nonequilibrium Green functions show that shift-current physics survives far beyond perturbative bulk formulas. In Rice-Mele chains attached to metallic electrodes and driven by femtosecond pulses, the dc photocurrent appears as nonadiabatic quantum pumping in the presence of left-right asymmetry. The transport from the illuminated region is superballistic, with displacement $q=0$1 and fitted exponents $q=0$2 in clean chains and $q=0$3 in diffusive chains, and the dc response persists at subgap frequencies through two-photon processes with $q=0$4 (Bajpai et al., 2018).

First-principles computation has accordingly diversified. Localized Gaussian-basis implementations reproduce the length-gauge and velocity-gauge shift conductivities while folding Brillouin-zone sums to the irreducible zone through crystalline and magnetic symmetry. Benchmarks on monolayer MoS$q=0$5, monolayer GeS, GaAs, and BaTiO$q=0$6 show near-identical spectra in both gauges when sufficiently complete Gaussian bases are used (García-Blázquez et al., 2023). In parallel, Wannier-projector implementations evaluate the quantum Hermitian connection and related tensors directly from projector derivatives corrected by dipole and quadrupole Wannier matrices, yielding shift-current spectra in GeS that agree with sum-rule and Wilson-loop approaches while preserving component-wise gauge invariance (Guo et al., 11 Sep 2025).

These model and computational developments also broaden the conceptual domain of the subject. The shift current is not exhausted by the textbook noncentrosymmetric two-band bulk picture; flat-band momentum shifts, open-device pumping, degeneracy-safe projector methods, and explicit multiband pathways have become standard parts of the modern description (Kaplan et al., 2021, Bajpai et al., 2018).

6. Experimental signatures, emergent functionalities, and scope

Magnetic systems now provide direct access to geometric channels that are absent in conventional nonmagnetic bulk photovoltaic experiments. In the van der Waals antiferromagnet MnPSe$q=0$7, a linear injection current and a circular shift current were observed and shown to reverse sign when the Néel vector was flipped. The helicity-odd circular component onsets at $q=0$8 K, while the ratio of the Fourier-transformed shift and injection signals yields an injection lifetime $q=0$9 fs, confirming the distinct instantaneous versus relaxing dynamics expected for shift and injection channels (Tian et al., 21 May 2026).

The same quantum-geometric logic extends to the lattice sector. Rectified Raman forces can be decomposed into phononic injection and shift forces, the direct phononic analogues of photogalvanic injection and shift currents. For linearly polarized light the resonant coefficients are

$$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$0

so that the injection force is governed by the quantum metric and the electron-phonon force asymmetry $$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$1, while the shift force is governed by the quantum metric and a phononic shift vector $$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$2. In the bilayer Haldane model, the resonant shift force is impulsive and requires time-reversal breaking (Pimlott et al., 23 Jul 2025).

Quantum-geometric shift current has also become a metrological resource. In an exciton-polariton detector with diamagnetic coupling $$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$3, the time-integrated shift current obeys the exact sum rule

$$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$4

while the integrated shot-noise Fano factor is

$$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$5

For pure photonic states, this yields

$$A^{b}_{n,m;a}=\partial_{k^a}A^{b}_{n,m}-i\left(A^{a}_{n,n}-A^{a}_{m,m}\right)A^{b}_{n,m},$$6

so the shift-current shot noise directly encodes the quantum Fisher information density of the incident light. Numerical demonstrations were given for optical Schrödinger cat states and squeezed vacuum states (Barts et al., 31 Mar 2026).

Taken together, these results establish quantum-geometric shift current as a broad organizing principle rather than a single bulk optical formula. The common element is the gauge-invariant displacement associated with a quantum transition—optical, scattering-induced, finite-momentum, magnetic, phononic, or static-bias-driven—and the common language is the geometry of projectors, Berry connections, shift vectors, quantum metrics, and higher geometric connections. A persistent theme across the modern literature is that the conventional real-space shift picture remains foundational, but is no longer sufficient on its own: multistate geometry, quantum-metric dipoles, geometric torsion, and momentum-space displacement have become equally central parts of the subject (Avdoshkin et al., 2024, Kitamura et al., 2 Jul 2026).