Shot-Noise Anisotropy: Concepts & Applications

• Shot-noise anisotropy is defined as the directional dependence of noise fluctuations arising from discrete events, observable in transport and astrophysical systems.
• It manifests in varied experimental contexts such as spintronic molecular junctions and gravitational-wave backgrounds, where anisotropic contributions influence current correlations and covariance structures.
• Measurement strategies leverage spectral analysis and interference patterns to extract anisotropic signatures, providing insights into energy transport and source discreteness.

Shot-noise anisotropy denotes different, domain-specific phenomena linked by a common theme: fluctuations generated by discreteness acquire a directional, orientational, or multipolar structure. In spintronic and molecular-scale transport, it refers to the dependence of shot noise on the orientation of magnetic moments and anisotropy axes, or on anisotropy-controlled spin dynamics. In angular clustering and gravitational-wave analyses, the shot-noise term in the power spectrum itself can be isotropic and fixed, while the shot-noise contribution to covariance, or the realization-level anisotropy produced by a finite population of discrete sources, is anisotropic in multipole space or on the sky. In ultrafast spectroscopy, closely related methodology addresses anisotropy measurements under shot-to-shot fluctuations by alternating polarization conditions on a per-shot basis (Filipovic, 2024, Tessore et al., 4 Jul 2025, Cusin et al., 24 Feb 2025).

1. Conceptual scope and formal definitions

In a broader sense, shot-noise anisotropy refers to the dependence of shot noise on the orientation of magnetic moments and anisotropy axes in a spintronic system (Filipovic, 2024). In that usage, anisotropy is dynamical and internal: a preferred axis, a tilt angle, or anisotropic exchange modifies transport channels, spin flips, and current correlations.

A different usage arises for discrete tracers on the sphere. For angular clustering power spectra, the observed spectrum can be written as

$$C_\ell^{AB} = \frac{1}{4\pi}\sum_{i i'} P_\ell(\cos\theta_{ii'}) ,$$

and splitting the sum into true pairs $i\neq i'$ and degenerate pairs $i=i'$ gives

$$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$

Here the additive contribution from degenerate pairs is the shot-noise term, whereas $\tilde C_\ell^{AB}$ is the clustering signal carried by distinct pairs. In this formulation, the shot-noise term in the power spectrum itself is isotropic and fixed, but the shot-noise contribution to the covariance is generically anisotropic and non-diagonal in multipole space (Tessore et al., 4 Jul 2025).

For gravitational-wave backgrounds the same discreteness logic is reformulated in terms of source populations. In the nano-Hz background from supermassive black hole binaries, the angular correlation of the fractional strain-squared field contains a Dirac-delta shot-noise term,

$$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$

and for $\ell>0$,

$$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$

This defines an effective number of sources $N_{\rm eff}$: fewer effective sources imply larger anisotropy (Lin et al., 18 Feb 2026).

These usages are compatible rather than contradictory. They distinguish anisotropy of a shot-noise observable in transport, anisotropy of covariance induced by discrete sampling, and anisotropy of a stochastic background generated by a finite population of discrete sources.

2. Molecular junctions and anisotropy-controlled shot noise

A concrete realization is a junction formed by a single electronic level coupled to two metallic leads and exchange-coupled to a large molecular spin $\vec S(t)$ that precesses in a static magnetic field $i\neq i'$0. The molecular spin has uniaxial magnetic anisotropy with axis along $i\neq i'$1, described by

$i\neq i'$2

Treating the molecular spin classically and neglecting backaction and damping, its precession frequency is modified to

$i\neq i'$3

The anisotropy parameter $i\neq i'$4 therefore controls the magnitude of the precession frequency and, for sufficiently large $i\neq i'$5, its sign. When $i\neq i'$6, the precession direction reverses relative to Larmor precession (Filipovic, 2024).

Because the exchange field is periodic in time, the orbital Hamiltonian is periodic and admits a Floquet-like treatment. The quasienergy spectrum contains four quasienergies $i\neq i'$7, with pairs connected by $i\neq i'$8 and $i\neq i'$9. Elastic tunnelling proceeds through a given quasienergy without energy exchange, whereas inelastic tunnelling involves absorption or emission of $i=i'$0 with a concomitant spin flip. In transport, steps in $i=i'$1 and in shot noise appear when a chemical potential crosses a quasienergy $i=i'$2, while peak–dip or dip–peak structures arise when elastic and inelastic pathways interfere destructively.

The central shot-noise observable is the zero-frequency autocorrelation $i=i'$3, together with the Fano factor

$i=i'$4

Anisotropy affects shot noise through the precession frequency $i=i'$5, the quasienergy spectrum $i=i'$6, and the mixing amplitude $i=i'$7. The dependence on orientation is sharply constrained. For $i=i'$8, $i=i'$9, so there is no spin-precession effect and the noise is independent of $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$0. For $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$1, $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$2 and the anisotropy term $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$3 vanishes, so shot noise is again independent of $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$4. Thus anisotropy in noise appears only when both transverse and longitudinal components are nonzero, $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$5 and $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$6, i.e. $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$7.

At zero bias and zero temperature, only spin-pump-induced shot noise remains. In this regime the noise is suppressed as $$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$8 is reduced by anisotropy, and when

$$C_\ell^{AB} = \tilde C_\ell^{AB} + \frac{N_{AB}}{4\pi}.$$9

precession stops, the precession-driven inelastic channels close, and the shot noise at zero bias drops to zero. A plausible implication is that uniaxial anisotropy acts as an internal control parameter for switching off spin-pump noise without changing the external bias.

3. Anisotropic exchange, field angle, and higher cumulants

A second transport meaning of shot-noise anisotropy appears in helical edge channels and in single-molecule magnets, where anisotropic spin couplings reshape current fluctuations and higher cumulants. For a helical edge of a two-dimensional topological insulator coupled to a local magnetic impurity with spin $\tilde C_\ell^{AB}$0, the exchange Hamiltonian contains anisotropic terms $\tilde C_\ell^{AB}$1, $\tilde C_\ell^{AB}$2, and $\tilde C_\ell^{AB}$3 in addition to the rotationally symmetric $\tilde C_\ell^{AB}$4 term. The paper characterizes the backscattered-current Fano factor as

$\tilde C_\ell^{AB}$5

With fully generic anisotropic coupling, $\tilde C_\ell^{AB}$6 depends nontrivially on $\tilde C_\ell^{AB}$7, satisfies $\tilde C_\ell^{AB}$8, and reaches the super-Poissonian limit $\tilde C_\ell^{AB}$9 for $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$0, reflecting correlated pairs of backscattering events. At high frequencies, the correlations are cut off and the ratio tends to $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$1 (Nagaev et al., 2018).

For single-molecule magnets above the sequential tunneling threshold, the anisotropy is geometric: the zero-frequency current noise and higher cumulants depend on the angle $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$2 between the applied magnetic field and the SMM easy axis, and on the transverse anisotropy $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$3. In the absence of the small transverse anisotropy, when $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$4, the maximum peak of shot noise first increases and then decreases with increasing $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$5 from $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$6 to $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$7. When a small transverse anisotropy is included, the shot noise can reach up to super-Poissonian value from sub-Poissonian value. For $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$8, the maximum peaks of the shot noise and skewness can be reduced from a super-Poissonian to a sub-Poissonian value with increasing $$\omega_{h^2}(\cos\vartheta) = \omega_{h^2}^{\rm LSS}(\cos\vartheta) + \frac{\langle h^4\rangle}{\langle h^2\rangle^2}\, \frac{\delta_D(1-\cos\vartheta)}{2\pi},$$9 from $\ell>0$0 to $\ell>0$1, and the super-Poissonian behavior of the skewness is more sensitive to the small $\ell>0$2 than shot noise (Xue et al., 2010).

In both systems, the microscopic mechanism is the competition between fast and slow transport channels. This suggests a unifying interpretation for transport realizations of shot-noise anisotropy: anisotropy reorganizes state occupations and transition rates, thereby changing the temporal bunching of tunneling events more strongly than it changes the mean current.

4. Discrete tracers, isotropic shot-noise terms, and anisotropic covariance

For angular clustering of discrete points on the sphere, the central conceptual result is negative: the true shot-noise contribution cannot have a non-Poissonian value, even though all point processes with non-trivial two-point statistics are non-Poissonian (Tessore et al., 4 Jul 2025). The additive term in the power spectrum is fixed by degenerate pairs,

$\ell>0$3

or $\ell>0$4 in the homogeneous auto-spectrum limit. Apparent deviations from a naive Poisson plateau arise when significant correlations or anti-correlations are localised on small spatial scales. However, such deviations always correspond to a physical difference in two-point statistics, not a difference in noise.

The anisotropy enters at the covariance level. For shot-noise-subtracted spectra $\ell>0$5, the two-point piece in the product $\ell>0$6 can be written as

$\ell>0$7

with

$\ell>0$8

Even for a Gaussian density field with Poisson sampling, this induces a non-diagonal covariance in $\ell>0$9. The usual Gaussian covariance appears as the monopole approximation; the full covariance contains additional off-diagonal contributions.

A closely related formalism appears in pulsar timing arrays when the gravitational-wave background is generated by a finite number $$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$0 of discrete sources at random sky locations. Encoding the source distribution by a masking function $$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$1, the angular power spectrum of the masking function is

$$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$2

or $$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$3 if the number of sources is Poisson with mean $$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$4. The mean Hellings–Downs correlation is unchanged in shape, but the cosmic covariance acquires an additional term proportional to $$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$5,

$$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$6

For conventional PTA sources, the effects of shot noise are much larger than the effects of clustering (Allen et al., 2024).

Taken together, these results establish a precise distinction. In discrete-tracer power spectra, the additive shot-noise term is isotropic and fixed; anisotropy emerges through covariance structure, mode coupling, or realization-level fluctuations generated by a finite population.

5. Gravitational-wave backgrounds: white shot noise, strong frequency dependence, and cross-correlations

In the nano-Hz gravitational-wave background from supermassive black hole binaries, shot-noise anisotropy is fundamentally controlled by

$$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$7

and for $$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$8 the shot-noise power spectrum is white,

$$C_{\ell,h^2}^{\rm SN} = \frac{\langle h^4\rangle}{\langle h^2\rangle^2} = \frac{4\pi}{N_{\rm eff}}.$$9

For GW-driven circular binaries, the paper finds

$N_{\rm eff}$0

so the shot-noise anisotropy rises steeply towards higher frequencies because binaries spend less time per logarithmic frequency interval and fewer effective sources contribute. The mean shot-noise anisotropy typically lies close to or above the broad frequency-band NANOGrav upper limits, whereas the LSS-induced anisotropies are at least two to three orders of magnitude smaller (Lin et al., 18 Feb 2026).

In the 10–100 Hz astrophysical gravitational-wave background, shot noise caused by the finite number of compact-binary coalescences contributes an $N_{\rm eff}$1-independent offset $N_{\rm eff}$2 to the measured angular spectrum, so that

$N_{\rm eff}$3

A method based on combining statistically-independent data segments constructs cross-spectra between different segments,

$N_{\rm eff}$4

and averages them to obtain an unbiased estimate of the true astrophysical spectrum, removing the offset due to shot noise power. In the limit of many data segments, the variance approaches the Cramér–Rao bound (Jenkins et al., 2019).

Cross-correlation with galaxy surveys plays a parallel role in both the ground-based and PTA bands. In the 10–100 Hz band, the spatial and temporal discreteness of sources leads to shot noise that may swamp attempts at measuring anisotropy, but cross-correlating the gravitational wave background and a galaxy catalog improves the chances of a first detection of the background anisotropy, given sufficient sensitivity (Alonso et al., 2020). In the PTA band, the spatial discreteness of supermassive black hole binaries introduces shot noise which, in certain regimes, would overwhelm efforts to measure anisotropy; cross-correlating a gravitational wave background map with a sufficiently dense galaxy survey mitigates this issue and improves by more than one order of magnitude the prospects for a first detection of the background anisotropy. With a futuristic scenario with an effective number of frequencies equal to $N_{\rm eff}$5, the detection of the spectral amplitude can be achieved combining the first $N_{\rm eff}$6 multipoles, with a threshold to resolve single events SKA-like (Cusin et al., 24 Feb 2025).

A plausible synthesis is that gravitational-wave applications realize both meanings of shot-noise anisotropy at once: a white, isotropic shot-noise spectrum in harmonic space and a strongly anisotropic phenomenology in detectability, covariance, and frequency dependence.

6. Measurement strategies, experimental inference, and recurrent misconceptions

In transport systems, anisotropy can be inferred from current and noise characteristics. In the molecular-junction problem, steps and interference-related features at specific biases correspond to $N_{\rm eff}$7, and because $N_{\rm eff}$8 have known analytical forms, one can extract $N_{\rm eff}$9, $\vec S(t)$0, $\vec S(t)$1, and $\vec S(t)$2. Zero-bias noise versus magnetic field $\vec S(t)$3 shows zeros where $\vec S(t)$4, providing a direct handle on $\vec S(t)$5 or $\vec S(t)$6 (Filipovic, 2024). In helical edges and single-molecule magnets, the dependence of the Fano factor and skewness on anisotropic couplings or on the angle between field and easy axis supplies a probe of microscopic scattering structure (Nagaev et al., 2018, Xue et al., 2010).

In ultrafast spectroscopy, the methodological problem is different but adjacent. Transient absorption anisotropy is usually retrieved from two consecutive transient-absorption measurements with parallel and perpendicular pump–probe polarization, so even minor systematic errors lead to drastic changes in the anisotropy signal. Alternating shot-to-shot detection of TA measurements with different pump polarization minimizes systematic errors due to laser fluctuations. The adopted formula averages first and then forms the anisotropy ratio,

$\vec S(t)$7

because the conceptually simple per-shot formula is very noisy (Binzer et al., 18 Mar 2025).

The main misconception addressed across fields concerns what is and is not anisotropic. In angular clustering power spectra, the shot-noise term from degenerate pairs is isotropic and fixed; what is often described as non-Poissonian shot noise is unresolved small-scale two-point signal (Tessore et al., 4 Jul 2025). In PTA and AGWB analyses, the source population can generate large anisotropy through discreteness even when the mean background is isotropic (Allen et al., 2024, Lin et al., 18 Feb 2026). In molecular transport, by contrast, anisotropy resides directly in the dependence of current fluctuations on a preferred axis, a tilt angle, or an anisotropic exchange tensor (Filipovic, 2024).

Shot-noise anisotropy is therefore not a single formal object but a family of precise phenomena tied to discreteness and directionality. Depending on the system, it may denote an anisotropic noise response in a transport observable, an anisotropic covariance generated by self-pairs and mode coupling, or a sky anisotropy produced by a finite population of discrete emitters.