Shift Current Shot Noise

• Shift current shot noise refers to the current fluctuations arising from the dc photocurrent generated by light-induced interband transitions, driven by the Berry connection in noncentrosymmetric materials.
• The two-band Floquet model and Keldysh Green’s functions show that while the average shift current is set by the geometric shift per transition, the noise is controlled by carrier dynamics and velocity differences post-excitation.
• Extensions using exciton-polariton frameworks link shot noise to photon-number variance and quantum Fisher information, offering a novel approach to probe quantum states and photodetection mechanisms.

Shift current shot noise denotes the current fluctuations associated with a shift current, a dc photocurrent generated in noncentrosymmetric systems under illumination without any external dc electric field. In the solid-state photovoltaic setting, the shift current is a geometric interband response rooted in the Berry connection of Bloch wavefunctions, whereas the bias-dependent correction to the current and the nonequilibrium current noise are controlled by the difference of group velocities between the photoexcited bands and the relaxation time (Morimoto et al., 2018). In a later exciton-polariton formulation, the shot noise of a nonlinear, quantum-geometric shift current was proposed as a detector of photon-number fluctuations and quantum Fisher information (QFI), because the integrated current depends only on the mean photon number while the noise depends on the photon-number variance (Barts et al., 31 Mar 2026).

1. Geometric origin of the shift current

A shift current is a dc photocurrent generated in a noncentrosymmetric crystal under illumination without any external dc electric field. Unlike conventional photovoltaic currents in p–n junctions, it does not rely on built-in fields, carrier drift, or diffusion of photocarriers toward electrodes. Its microscopic origin is the Berry connection of Bloch wavefunctions. For a Bloch state $\lvert \psi_n(k)\rangle = e^{ikx}\lvert nk\rangle$, the intracell coordinate is

$$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$

where $a_n(k)$ is the Berry connection of band $n$. When light induces an interband transition $1\to 2$, the electron’s intracell position changes; in a crystal with broken inversion symmetry $\mathcal I$, this shift has a preferred direction and produces a dc current (Morimoto et al., 2018).

The central geometric quantity is the shift vector

$$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$

This expression makes explicit that the photocurrent is controlled by interband matrix elements and Berry-connection differences rather than by semiclassical drift. The zero-bias shift current is

$$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$

or equivalently

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$

with

$$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$

These expressions contain only interband optical matrix elements and the shift vector; crucially, they do not contain the band velocity difference $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$0 nor $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$1 (Morimoto et al., 2018).

This separation between geometric interband response and ordinary transport is the defining conceptual feature of shift current shot noise. The average current is set by the deterministic displacement associated with each optical transition, while fluctuations arise from the stochastic population and motion of photoexcited carriers after the transition. In the later exciton-polariton framework, the same geometric idea reappears through a diamagnetic coupling $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$2 that “originally represents the interband diamagnetic interaction, which is related to the shift vector” (Barts et al., 31 Mar 2026).

2. Current–voltage relation and noise in shift-current photovoltaics

The 2018 theory combines a two-band Floquet model for a monochromatically driven insulator, Keldysh nonequilibrium Green’s functions to treat the steady state under light irradiation, and a gauge-invariant gradient expansion in a weak dc electric field $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$3. The resonant Floquet subspace consists of the valence band plus one photon and the conduction band plus zero photons, with Hamiltonian

$$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$4

where $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$5. The driven steady state is stabilized by coupling to a bath modeled by a constant broadening $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$6, and the lesser and greater Green’s functions are written as

$$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$7

For the dc response, the Green’s function is expanded as

$$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$8

This yields the linearized $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$9-$a_n(k)$0 relation

$a_n(k)$1

with

$a_n(k)$2

and

$a_n(k)$3

Here $a_n(k)$4 and $a_n(k)$5 are the diagonal velocity matrix elements, $a_n(k)$6 is the velocity difference between the photoexcited electron and hole sectors, and $a_n(k)$7 is the relaxation time (Morimoto et al., 2018).

The zero-frequency current noise is defined in terms of a local current operator

$a_n(k)$8

and

$a_n(k)$9

Within the same Floquet two-band steady state, the principal result is

$n$0

This formula shows that the shot noise is proportional to $n$1 and vanishes when $n$2. The paper also emphasizes that the noise does not contain the usual equilibrium-type current-noise part proportional to the current, $n$3, so the usual Poissonian interpretation is inapplicable or at least highly nonstandard (Morimoto et al., 2018).

The distinction is therefore sharp. The average shift current is set by the real-space shift per optical transition and is essentially independent of band transport details. The bias response and the noise are set by the motion and lifetime of the created carriers after excitation. This suggests that a finite $n$4 can coexist with very small differential conductivity and very small shot noise.

3. Suppression mechanisms, flat bands, and Landau-level realizations

The separation between $n$5 and $n$6 immediately produces a materials-design principle. The 2018 paper states that one can, in principle, design a system with large shift current, small $n$7-$n$8 slope, and strongly suppressed shot noise by maximizing the geometric factor $n$9 while minimizing $1\to 2$0. Flat-band systems are singled out because the average shift current can remain finite even when transport channels are dispersionless (Morimoto et al., 2018).

Noncentrosymmetric two-dimensional systems with Landau levels are proposed as especially promising because Landau levels are effectively flat bands. For flat bands,

$1\to 2$1

so

$1\to 2$2

while $1\to 2$3 can remain finite. This is the ideal regime identified in the paper: finite photocurrent, no voltage dependence, and no nonequilibrium current noise (Morimoto et al., 2018).

For graphene Landau levels with inversion breaking, the model Hamiltonian is

$1\to 2$4

and for the $1\to 2$5 transition the response is

$1\to 2$6

This vanishes unless inversion symmetry is broken so that the valley contributions do not cancel. For the surface states of a 3D topological insulator with trigonal warping,

$1\to 2$7

the nonlinear conductivity near the $1\to 2$8 Landau-level transition is

$1\to 2$9

The paper further notes that two interband transitions with the same resonance frequency can contribute with opposite signs to shift current, unlike linear absorption where they add (Morimoto et al., 2018).

The same work estimates 2D nonlinear conductivities around $\mathcal I$0 for graphene, $\mathcal I$1 for MoS$\mathcal I$2, and $\mathcal I$3 for TI surface states, larger than a typical 2D shift-current material such as monolayer GeS ($\mathcal I$4). For a TI sample of size $\mathcal I$5, light intensity $\mathcal I$6, and $\mathcal I$7, the estimated shift current is of order $\mathcal I$8. The maximum dc field is limited by the resonance condition,

$\mathcal I$9

with $$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$0 for Landau levels, and the monochromatic conversion efficiency is estimated as

$$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$1

For $$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$2 and $$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$3, the quoted efficiencies are approximately $$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$4 for graphene, $$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$5 for MoS$$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$6, and $$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$7 for TI surface states (Morimoto et al., 2018).

4. Exciton-polariton shift current and neutral-excitation photodetection

A distinct theoretical development reformulates shift current shot noise as a probe of many-photon quantum states. The detector consists of an exciton-polariton system in a cavity, where a single cavity photon mode hybridizes with a neutral exciton mode. The minimal Hamiltonian is

$$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$8

To generate the shift current, a static probe field $$R= \operatorname{Im}\!\left[\frac{(\partial_k v)_{12}}{v_{12}}\right] = \operatorname{Im}[\partial_k (\log v_{12})] + a_1-a_2 .$$9 is included through

$$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$0

so the current operator is

$$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$1

The coupling $$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$2 is stated to be related to the shift vector, and time-reversal symmetry fixes the relative phase of $$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$3 and $$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$4, with $$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$5 chosen real and $$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$6 purely imaginary (Barts et al., 31 Mar 2026).

The key physical point is that the exciton and photon are neutral, so the current is not due to photocarrier transport. Instead, a photon excites a neutral exciton-polariton degree of freedom, and the nonlinear geometric coupling converts that optical excitation into a charge shift. The paper states that “In all these cases, no photocurrent shot noise occurs from photocarriers and hence the current noise is expected to reflect the quantum nature of the photons” (Barts et al., 31 Mar 2026).

Dissipation is essential. The exciton is coupled to an external bosonic bath, and under the Markov approximation with negligible bath occupation the reduced density matrix obeys

$$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$7

The initial photonic state is expanded in the Glauber-Sudarshan representation,

$$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$8

For arbitrary initial photon state, the mean current takes the form

$$J_{\rm shift} = \frac{2\pi e^3}{\hbar^2 \omega^2} |E(\omega)|^2 \int [dk]\, \operatorname{Im}\!\left[\left(\frac{\partial v}{\partial k}\right)_{12} v_{21}\right] \delta(\omega_{21}-\omega),$$9

and the integrated current is

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$0

where

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$1

This result is also obtained from the exact continuity relation

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$2

Under the model assumptions and integration over all time, the total shift charge therefore depends only on the mean initial photon number (Barts et al., 31 Mar 2026).

5. Fano factor, photon-number fluctuations, and QFI

The connected current-current correlator in the exciton-polariton setting is

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$3

and the Fano factor is defined by

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$4

Using the continuity relation inside the two-time correlator, the paper derives the exact sum rule

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$5

The integrated current is therefore insensitive to the detailed quantum state of the incident light, whereas the noise retains the photon-number variance. This is the precise sense in which the detector is “blind” at the level of the first cumulant but “remembers” the quantum information in the second cumulant (Barts et al., 31 Mar 2026).

For phase estimation generated by the photon-number operator $$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$6, the paper uses

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$7

for pure states and defines the QFI density

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$8

Hence

$$J_{\rm shift} = \sigma^{(2)}(\omega) |E(\omega)|^2,$$9

The paper mostly discusses this equality in the pure-state examples it studies and notes that it does not develop a separate mixed-state QFI formula in detail (Barts et al., 31 Mar 2026).

Three examples are analyzed numerically by solving the full Lindblad equation in a truncated Fock space of up to 15 total particles. For a coherent state $$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$0, the photon statistics are Poissonian, $$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$1, so

$$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$2

For the even optical Schrödinger cat state,

$$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$3

the QFI density is

$$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$4

and therefore $$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$5. For the squeezed vacuum state,

$$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$6

with $$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$7, the QFI density is

$$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$8

so

$$\sigma^{(2)}(\omega) = \frac{2\pi e^3}{\hbar^2 \omega^2} \int [dk]\, |v_{12}|^2 R\, \delta(\omega_{21}-\omega).$$9

The numerical results show that the average current does not distinguish the cat state from any other state with the same mean photon number, but the noise does; similarly, the Fano factors separate according to the QFI of the states (Barts et al., 31 Mar 2026).

6. Conceptual boundaries, assumptions, and common misunderstandings

A recurring misunderstanding is to treat shift current shot noise as an ordinary extension of Poissonian transport noise. The 2018 analysis explicitly rejects this identification: the zero-bias shift current is not carried by noisy drift of long-lived carriers, and there is no conventional $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$00 term. In conventional photodiodes, light creates electron-hole pairs, built-in fields separate them, carriers drift or diffuse to contacts, and current and noise are both tied to stochastic carrier transport. In a shift-current photovoltaic, no built-in junction is needed, broken inversion symmetry sets the current direction, and the dc current appears at zero bias from the geometric shift during optical excitation. Noise is suppressed when the post-excitation carrier velocities are equal or vanish (Morimoto et al., 2018).

Both formalisms rely on strong idealizations. In the solid-state photovoltaic theory, the assumptions include a two-band Floquet truncation, noninteracting electrons, a simple bath self-energy with constant broadening $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$01, zero temperature or insulating initial state, a large-electrode approximation used to neglect one noise term, and the treatment of only electronic current noise. The paper explicitly notes that for photodetection of weak light one should also include photon shot noise and quantum statistics of photons, which are left for future work. It also states that the relation to experimentally measured terminal noise in a full device is not developed in detail (Morimoto et al., 2018).

The exciton-polariton theory introduces a different set of assumptions: a Markov bath, zero effective temperature, a single photon mode plus a single exciton mode, the bosonic exciton approximation, normal ordering and coherent-state decomposition for arbitrary initial photonic states, and dissipation only through the exciton channel. Finite $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$02 is essential to regularize the time integrals and convert polarization into net current, yet the integrated shift charge $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$03 is independent of $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$04 once integrated to long times; $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$05 is a singular limit. The paper also notes a subtlety in the two-time correlator: the superoperator construction misses certain interference terms caused by unequal-time commutators, producing a small $$x_n = a_n(k) = -i\langle nk \vert \nabla_k \vert nk \rangle ,$$06-oscillatory offset in the detailed time dependence, but the exact sum rule still fixes the integrated real part, so the Fano factor remains exact (Barts et al., 31 Mar 2026).

Taken together, these two lines of work define shift current shot noise as a subject at the intersection of nonlinear quantum geometry, nonequilibrium transport, and photodetection theory. In one setting it explains why a finite geometric photocurrent can coexist with unusually low nonequilibrium current noise; in another it converts photon-number fluctuations of nonclassical light into measurable electrical shot noise. A plausible implication is that shift current shot noise provides a framework in which the average current and its fluctuations are not merely different observables, but diagnostics of different physics: geometric charge transfer in the first cumulant and stochastic transport or photon-number statistics in the second.