Diagnosing electronic phases of matter using photonic correlation functions

TL;DR

This work introduces photonic correlation spectroscopy as a direct probe of electronic correlations in strongly interacting materials, linking scattered-photon correlators to spin and charge dynamics via input–output theory and a multi-channel T-matrix formalism. By employing frequency filters and homodyne schemes, the authors map G^(1), G^(2), and quadrature observables to a hierarchy of matter operators (R^(1), R^(2)) that encode spin, charge, and topology, enabling sector-resolved measurements in a Mott insulator described by the single-band Fermi–Hubbard model at half-filling. The paper presents concrete applications: measuring static spin chirality on kagome/triangular lattices, diagnosing mixed spin–charge dynamics, assessing magnon contributions, and detecting fractional statistics through conditional and connected photonic correlators, with clear experimental feasibility in 2D materials and moiré systems. The approach broadens the experimental toolkit beyond linear response, offering a pathway to uncover hidden many-body states, including chiral spin liquids and anyonic excitations, by analyzing quantum properties of scattered light.

Abstract

In the past couple of decades, there have been significant advances in measuring quantum properties of light, such as quadratures of squeezed light and single-photon counting. Here, we explore whether such tools can be leveraged to probe electronic correlations in the many-body quantum regime. Specifically, we show that it is possible to probe certain spin, charge, and topological orders in an electronic system by measuring the correlation functions of scattered photons. We construct a mapping from the correlators of the scattered photons to those of a correlated insulator, particularly for Mott insulators described by a single-band Fermi-Hubbard model at half-filling. We show that frequency filtering before photodetection plays a crucial role in determining this mapping. We find that if the ground state of the insulator is a gapped spin liquid, a photon-pair correlation function, i.e., $G^{(2)}$, can detect the presence of anyonic excitations with fractional mutual statistics. Moreover, we show that correlations between electromagnetic quadratures can be used to detect expectation values of static spin chirality operators on both the kagome and triangular lattices, thus being useful in detecting chiral spin liquids. More generally, we show that a series of hitherto unmeasured spin-spin and spin-charge correlation functions of the material can be extracted from photonic correlations. This work opens up access to probe correlated materials, beyond the linear response paradigm, by detecting quantum properties of scattered light.

Figures (20)

• Figure 1: A system of itinerant electrons is irradiated with a laser. Conventionally, photodetectors measure the intensity of the scattered photons, and the correlations are ignored. In this paper, we propose a Hanbury Brown-Twiss-type setup to measure correlations between pairs of photons. We allow for frequency filters, $\mathcal{F}_1$ and $\mathcal{F}_2$, before detection, and a time delay, $\tau$, between detection events.
• Figure 2: Schematic illustration of different photon scattering processes. (a-c) The blue- and orange-shaded regions represent the lower and higher energy sectors, respectively, separated by an optical gap of order $U$. For the Fermi-Hubbard model at half-filling, $U$ corresponds to the on-site repulsion, with the low- and high-energy sectors identified as spin and charge sectors, respectively. More generally, the presented formalism applies to any insulator with an optical gap. The laser frequency $\omega_{L}$ is assumed to be detuned from $U$. The three terms in $\hat{M}_{\lambda}$, defined in Eq. \ref{['eq:Meffint']} correspond to different pathways leading to emission of a photon. These three pathways are depicted schematically as follows: (a): Raman process -- absorption of a laser photon followed by the emission of a photon with a frequency near $\omega_{L}$. This process is governed by the effective matter operator $\hat{A}_{\lambda}$, which mediates transitions from the state $\ket{I}$ to $\ket{J}$ within the same (lower energy) sector. (b): Absorption of two photons followed by emission of a photon of frequency near $2\omega_{L}-U$. This process involves the effective matter operator $\hat{B}_{\lambda}$, which transitions the state $\ket{I}$ in the lower energy sector to $\ket{K}$ in the higher energy sector. (c): Emission of a photon of frequency near $U$ originating from a state $\ket{K}$ in the higher energy sector that was previously accessed via process (b). This emission is mediated by the effective matter operator $\hat{C}_{\lambda}$, which transitions the state $\ket{K}$ to $\ket{F}$ in the lower energy sector. Panels (d-f) illustrate the corresponding intensity versus frequency profiles. Notably, emission into sideband (e) is necessarily accompanied by simultaneous emission into sideband (f).
• Figure 3: Schematic representation of the scattering process. (a) The initial state in the asymptotic past (at time $t=-T/2$, in the limit where it approaches $-\infty$), $\ket{\Psi(-T/2)}$ consists of the electromagnetic field in a laser-produced wavepacket state far away from the material. The material is assumed to be in an energy eigenstate $\ket{I}$. Around time $t=0$, the wavepacket spatially overlaps with the material and interacts with it for a duration proportional to the length of the wavepacket, which we assume to be much larger than its central wavelength (b) At $t=T/2$, in the asymptotic future, light and the material are entangled with each other and the resulting superposition is schematically depicted in the figure. The expansion depicted here is in the number of photon modes in the final state. The first term corresponds to the elastic scattering of light. The second set of terms corresponds to the emission of a photon in mode $\lambda$, leaving the material in a state $\hat{R}_{\lambda}^{(1)}\ket{I}$, where $\hat{R}_{\lambda}^{(1)}$ is an operator acting purely in the matter sector. For brevity, we have left out energy-conserving $\delta$-functions in the above schematic (for a more careful treatment, see Eq. \ref{['eq:outforcoherent']}). The third set corresponds to the emission of a pair of photons in modes $\lambda_{1}$ and $\lambda_{2}$. Therefore, correlation functions of photons map to correlation functions of matter operators like $\hat{R}^{(1)}_\lambda$ and $\hat{R}^{(2)}_{\lambda_1,\lambda_2}$. In this paper, we present a formalism to derive expressions for these matter correlation functions.
• Figure 4: Optical scheme for the measurement of the phase-sensitive second-order quadrature correlations between a pair of photons scattered off the material Eq. (\ref{['eq:X2def']}). One of the photons (shown as blue) is subjected to an additional (retarded) time delay $\tau$. After both photons pass through the respective frequency filters $\mathcal{F}_{j}$, each photon is mixed with a strong field of a local oscillator (annihilation operator denoted as $\hat{a}_{\text{L.O.}}$) using a beamsplitter. In our work, we consider the frequency of the local oscillator to be equal to the drive frequency $\omega_L$. The phase difference of the local oscillator with respect to the drive laser can be tuned. If $\hat{a}_{\lambda_j}$ is a scattered photonic mode, then for each of the two beam-splitters, the mode through each of the two output arms, $+1$ and $-1$ is $\tfrac{1}{\sqrt{2}}\left(\hat{a}_{\lambda_j}\pm \hat{a}_{\text{L.O.}} \right)$ respectively. First, let us consider the output from just one of the filters $\mathcal{F}_j$. The difference $G^{(1)}_{d_j;+}-G^{(1)}_{d_j;-}$ between the two arms of the beam splitter is proportional to a quadrature measurement $\expval{\hat{a}_{d_j}}e^{i\theta}+\text{c.c.}$ Now, let us consider the output from both the filters $\mathcal{F}_1$ and $\mathcal{F}_2$. We show that by measuring $G^{(2)}$ correlations between the output arms of the beam-splitter and by taking a suitable linear combination [Eq. \ref{['eq:homodynetrick']}], one can measure phase-sensitive second-order quadrature correlations between the scattered photons $\expval{\hat{a}_{d_2}(\tau)\hat{a}_{d_1}(0)}$.
• Figure 5: Measurement scheme for $H_{d_1,d_2}(t,\tau)\equiv \langle\hat{a}^{\dagger}_{d_1}(0)\hat{a}^{\dagger}_{d_2}(t+\tau)\hat{a}_{d_2}(t)\hat{a}_{d_1}(0)\rangle_{\text{out}}+\text{c.c.}$, which can be thought of as a $G^{(1)}(\tau)$ measurement at detector $d_2$ conditioned on detecting a photon at $d_1$. The mode shown in green here is split into two paths using a beamsplitter, and one of the paths is given a time delay $\tau$ with respect to the other. The two paths are made to interfere, and two $G^{(2)}$ measurements are taken between $d_1$ and each arm of $d_2$. We show in Appendix \ref{['app:homodyne']} that the desired correlator can be obtained this way.