Intensity Correlation Measurement Protocol

• Intensity correlation measurement protocols are techniques that quantify the cross-correlation between intensity fluctuations at distinct frequencies or spatial locations, building on HBT experiments.
• They leverage resonant conditions and Fourier decomposition to extract higher-order cumulants, such as the fourth cumulant, enabling the study of nonclassical noise behavior.
• Practical implementations use dual detection branches with precise filtering and calibration to isolate minute non-Gaussian signals, aiding mesoscopic physics and quantum device characterization.

Intensity correlation measurement protocols are designed to quantify the statistical interdependence between intensity fluctuations measured at two or more points in space, time, frequency, or other degrees of freedom. Rooted in the seminal Hanbury Brown and Twiss (HBT) experiments, these protocols have evolved to probe both fundamental properties of quantum and classical light and non-Gaussian noise in mesoscopic conductors. They are essential for extracting information that is inaccessible to first- or second-order (mean and variance) observables, such as the fourth cumulant of current fluctuations, higher-order photon correlations, or nonclassical emission statistics.

1. Fundamental Principles and Theoretical Foundations

The canonical intensity correlation protocol measures the cross-correlation between intensity (or noise power) fluctuations in two branches, often realized at two nonoverlapping frequencies or spatial locations. If $P_1(t)$ and $P_2(t)$ are the instantaneous powers detected after suitable filtering and dc blocking, the relevant correlator is: $G_2 = \langle \delta P_1(t) \delta P_2(t) \rangle$ with $\delta P_k(t) = P_k(t) - \langle P_k \rangle$. This second-order intensity correlation is closely analogous to the original optical HBT measurement, but it is generalized in frequency, time, and physical context.

A critical result established by measurements on tunnel junctions under ac excitation is that $G_2$ is directly proportional to the fourth cumulant of the current fluctuations: $$G_2 \propto \langle\!\langle i(f_1) i(\epsilon - f_1) i(f_2) i(-\epsilon - f_2) \rangle\!\rangle$$ where $i(f)$ is the Fourier component of the current and the double brackets denote the cumulant. This structure filters out the dominant Gaussian contribution (second moment) and isolates non-Gaussian "excess" fluctuations.

The normalized intensity correlation is defined as: $$g_2 = \frac{\langle P_1 P_2 \rangle}{\langle P_1 \rangle \langle P_2 \rangle} = 1 + \frac{G_2}{\langle P_1 \rangle \langle P_2 \rangle}$$ A value $g_2 > 1$ is interpreted as photon bunching and directly signals nontrivial correlations.

2. Protocol Architecture and Frequency Selection Rules

The protocol implementation typically utilizes two detection branches with nonoverlapping bandpass filters, selecting noise at frequencies $f_1$ and $f_2$. Each branch is followed by a microwave power detector, and a dc block ensures only instantaneous fluctuations are analyzed. In innovative mesoscopic setups, noise is excited by an ac voltage at frequency $f_0$ applied to the sample. The key result is that significant (nonzero) $G_2$ is observed only when the excitation and detection frequencies satisfy: $$f_0 = f_1 + f_2 \quad \text{or} \quad f_0 = |f_2 - f_1| \quad \text{or} \quad f_0 = \frac{|f_2 - f_1|}{2}$$ The specific resonance condition is set by the dc bias applied to the tunnel junction:

• At high dc bias, peaks in $G_2$ manifest at $f_0 = f_1 + f_2$ and $f_0 = |f_2 - f_1|$.
• At zero dc bias, $G_2$ peaks at $f_0 = (|f_2 - f_1|)/2$.

3. Fourier Decomposition and Analytical Formulation

The theoretical underpinning involves expanding the noise spectral density in harmonics of the ac excitation via: $$X_n = \int_{-\pi}^{\pi} S_0(V_{dc} + V_{ac} \cos \theta) e^{in\theta} \frac{d\theta}{2\pi}$$ where $S_0(V)$ is the unexcited noise spectral density. The correlator at the signal-matched excitation condition can be recast as: $G_2(f_0 = f_\pm/n) = K \cdot X_n^2$ with $K$ a normalization factor determined by detection bandwidths and frequency integration windows.

In the limit of small ac excitation $V_{ac}$, the lowest Fourier components dominate: $$X_1^2 \approx \frac{1}{4} \left(\frac{dS_0}{dV}\right)^2 V_{ac}^2$$

$$X_2^2 \approx \frac{1}{64} \left(\frac{d^2 S_0}{dV^2}\right)^2 V_{ac}^4$$

Thus, $G_2$ exhibits either quadratic or quartic scaling in $V_{ac}$, depending on whether the resonance is set by the first or second harmonic in the expansion.

4. Sensitivity to Non-Gaussian Fluctuations and Photon Statistics

By isolating the connected four-frequency current correlator, the protocol directly probes the fourth cumulant of current fluctuations, distinguishing non-Gaussian electronic noise. This is evidenced experimentally by observing photon bunching ($g_2 > 1$), which is absent in purely thermal or Poissonian emission. The positive $G_2$ observed in the measurement is a direct signature of nonclassical correlations in the photo–assisted shot noise of the tunnel junction.

The dependence of $G_2$ on both $V_{dc}$ and $V_{ac}$ allows precise control and selective amplification of desired harmonics. Tuning $V_{dc}$ changes which $X_n$ dominates, thereby enabling or suppressing correlation peaks.

5. Practical Considerations and Signal Extraction

The experimental realization of the described protocol requires:

• Two branches with well-defined nonoverlapping bandpass microwave filters centered at $f_1$ and $f_2$
• DC blocks in each branch to eliminate mean background and focus on fluctuations
• Power detectors capable of extracting the instantaneous noise power, with detection bandwidths $\Delta f_1$, $\Delta f_2$, and output integration over a frequency window $\Delta\epsilon$
• Synchronous acquisition and cross-correlation computation between branches to enable numerical evaluation of $G_2$

Noise powers are extracted as functions of time $P_k(t)$, and the fluctuating parts $\delta P_k(t)$ are used to build cross-correlation histograms. These are further analyzed in Fourier space to match the theoretical expressions for $G_2$. Calibration of bandwidths and careful correction for cross-talk, amplifier noise, and drift are necessary to access the low (but finite) non-Gaussian contributions.

6. Applications and Relevance to Mesoscopic and Quantum Devices

The protocol has significant implications for mesoscopic physics and quantum device characterization. Measuring $G_2$ provides direct access to the fourth cumulant, which describes the degree of non-Gaussianity in current fluctuations—critical for understanding electron–electron interactions, environment-induced effects, and fundamental quantum noise. The technique complements existing second-moment shot-noise measurements, offering a richer statistical probe. Practical applications include:

• On-chip microwave photon detectors
• Characterization of non-Gaussian noise in quantum conductors
• Probing photon statistics in regimes close to single-photon emission

The observed photon bunching links electronic shot noise in a mesoscopic tunnel junction to quantum-optical paradigms, deepening the conceptual connection between condensed matter physics and quantum optics.

7. Summary Table of Key Relations

Symbol/Relation	Physical Meaning
$G_2 = \langle \delta P_1 \delta P_2 \rangle$	Cross-correlation between intensity fluctuations
$G_2 \propto$ fourth cumulant	Sensitivity to non-Gaussian current statistics
$G_2(f_0 = f_\pm/n) = K X_n^2$	Resonant enhancement governed by Fourier components $X_n$
$$g_2 = 1 + (G_2 / \langle P_1 \rangle \langle P_2 \rangle)$$	Normalized intensity correlation
$$X_n = \int_{-\pi}^{\pi} S_0(V_{dc} + V_{ac} \cos\theta) e^{in\theta} \frac{d\theta}{2\pi}$$	Harmonic amplitudes in spectral noise expansion
$G_2 \propto X_1^2$ (high $V_{dc}$), $G_2 \propto X_2^2$ (zero $V_{dc}$)	Dominance of first or second harmonic depending on dc bias

This protocol exemplifies a direct, quantitative methodology for studying non-Gaussian fluctuations via intensity correlations in the frequency domain, establishing a powerful bridge between experimental methods in quantum optics and electronic transport in mesoscopic physics (Forgues et al., 2013).