⚛️ high-energy theory

Measuring correlations using local and nonlocal quenches

This paper proposes a theoretical method for measuring correlation functions, including non-Gaussian ones, in quantum field systems by analyzing how the system's average field evolution depends on its initial state following local or nonlocal quantum quenches.

Original authors: Alexey G. Mikhaylenko, Andrew G. Semenov

Published 2026-09-22

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CC BY 4.0

Original authors: Alexey G. Mikhaylenko, Andrew G. Semenov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the work of Alexey G. Mikhaylenko and Andrew G. Semenov, one must first grasp the nature of the quantum world they are studying. In physics, a system is not just a collection of particles sitting still; it is a dynamic field where values fluctuate and interact. The state of such a system is defined by its correlations, which describe how the behavior of one part of the system is linked to another. While simple systems follow predictable, bell-curve patterns known as Gaussian statistics, more complex systems exhibit non-Gaussian correlations, where the relationships between parts are far more intricate and difficult to map. For decades, scientists have struggled to measure these complex connections, especially in systems with many moving parts. The ability to map these hidden links is crucial for understanding how quantum systems evolve and how they might be used in future technologies, but the tools to see them have been limited.

In a theoretical proposal published in September 2026, researchers from the Lebedev Physical Institute and the Skolkovo Institute of Science and Technology suggest a new way to reveal these hidden connections. Their method relies on a concept called a "quantum quench." Imagine a calm pond representing a quantum system in its ground state. A quench is like dropping a stone into that pond, but instead of a physical rock, it is a sudden, brief disturbance applied to the system for a split second. The researchers propose that by carefully choosing how this disturbance is applied, they can force the system to reveal its internal correlations through the way it ripples afterward. By measuring the average movement of the field after this brief jolt, they can work backward to determine exactly what the correlations were before the jolt happened.

The paper explores two specific ways to deliver this jolt. The first is a "local quench," where the disturbance is focused on a single, tiny point in space. The researchers show that if you apply this type of push to a system, the way the field evolves afterward depends entirely on the correlations of the particles right at that spot. By repeating this process with different types of pushes, one can measure the average value of the field, the square of the field, the cube of the field, and so on. This allows scientists to build a complete picture of the correlations at that specific location, including the complex, non-Gaussian ones that are usually hard to detect. However, this method has a limitation: it only tells you about particles that are all clustered near the same point. It cannot easily tell you how a particle at point A is linked to a particle at point B if they are far apart.

To solve this problem, the authors introduce a second, more sophisticated method called a "nonlocal quench." In this scenario, the disturbance is applied simultaneously at several different points in space, rather than just one. The researchers demonstrate that this approach allows the system to reveal correlations between particles at these separate locations. By measuring the field's evolution in the immediate aftermath, and specifically looking at the area around one of the disturbance points before the ripples from the other points arrive, scientists can isolate and measure the link between those distant points. This effectively turns the system into a tool for mapping connections across space, not just within a single spot.

The team tested these ideas using mathematical models of scalar fields, which are simplified representations of quantum systems, considering both cases where the particles have mass and cases where they are massless. They found that the behavior of the system after the quench depends heavily on the dimension of space and whether the particles have mass. In a one-dimensional space with massless particles, the disturbance creates a ripple that settles into a constant value across the entire space, a unique behavior that persists indefinitely. In contrast, systems with massive particles show ripples that oscillate and fade over time, with the speed of the ripple front depending on how tightly the initial disturbance was focused. These findings provide a clear, analytical roadmap for how the average field behaves, confirming that the proposed method works in theory.

The researchers are careful to note that their current proposal applies to systems that do not have complex interactions between particles. They acknowledge that if the particles interact strongly with each other, the simple rules they derived would no longer hold, and more advanced techniques would be needed. Nevertheless, the paper establishes a solid foundation for a new kind of measurement. By using these controlled, short-lived disturbances, scientists could potentially perform a form of tomography on quantum states, reconstructing the full map of their correlations. This could open the door to studying the complex, non-Gaussian behaviors that define the most interesting and powerful quantum systems, offering a new lens through which to view the fundamental structure of matter.

Technical Summary: Measuring Correlations Using Local and Nonlocal Quenches

Problem Statement
The characterization of quantum many-body systems requires the measurement of correlation functions, including higher-order and non-Gaussian correlations. While experimental progress has enabled the measurement of various correlation functions, a specific theoretical procedure for extracting a broad class of these functions—particularly non-Gaussian ones—from the dynamics of a quantum field system remains an open challenge. Existing methods, such as local quench spectroscopy, primarily focus on dynamic properties or spectra rather than reconstructing the correlation structure of the initial state. This paper addresses the question of which correlation functions of an initial quantum state can be retrieved by observing the time evolution of observables following specific types of short-term perturbations (quantum quenches).

Methodology
The authors propose a theoretical framework based on quantum quenches, defined as perturbations acting for a very short duration, modeled by a Hamiltonian $\hat{H}_{\text{int}}(t) = \delta(t - t_Q)\hat{V}$ . This perturbation instantaneously transforms the system's density matrix $\hat{\rho}$ into $\hat{\rho}_Q = \hat{Q}\hat{\rho}\hat{Q}^\dagger$ , where $\hat{Q} = e^{-i\hat{V}/\hbar}$ .

The core methodology relies on the observation that the post-quench evolution of the average field $\langle \hat{\phi}(t, \vec{x}) \rangle_Q$ depends on the expectation values of the field and its conjugate momentum immediately after the quench. These post-quench values are, in turn, determined by the commutators of the perturbation operator $\hat{V}$ with the field operators, which encode the correlation functions of the initial state $\hat{\rho}$ .

The paper analyzes two distinct types of perturbations:

Local Quenches: The perturbation operator is localized in space near a single point $\vec{x}_Q$ , taking the form $\hat{V} = \lambda \hat{\phi}_S^n(\vec{x}_Q)$ , where $\hat{\phi}_S$ is a smeared field operator to avoid ultraviolet divergences.
Nonlocal Quenches: The perturbation operator involves products of field operators localized at different spatial points, $\hat{V} = \lambda \hat{\phi}_S(\vec{x}_{Q,1}) \dots \hat{\phi}_S(\vec{x}_{Q,n})$ .

The evolution of the average field is calculated using the retarded Green's function $G_R(t, \vec{x})$ of the free theory. The authors derive analytical expressions for the asymptotic behavior of the smeared retarded Green's function $G_R^S$ in both massive ( $M \neq 0$ ) and massless ( $M = 0$ ) scalar field theories across dimensions $d=1, 2, 3$ .

Key Contributions and Results

General Algorithm for Correlation Measurement: The paper establishes that by applying a quench with a specific operator power $n$ and measuring the subsequent evolution of the average field at the quench location, one can isolate and calculate the $n$ -point correlation functions of the initial state. Specifically, for a local quench $\hat{V} \propto \hat{\phi}^n$ , the deviation of the average field from its unperturbed evolution is proportional to the $(n-1)$ -point correlation function $\langle \hat{\phi}^{n-1} \rangle$ .
Local Quench Limitations and Capabilities:
- Local quenches allow for the sequential measurement of correlation functions of the form $\langle \hat{\phi}_S^\alpha(\vec{x}_Q) \rangle$ .
- The authors derive the asymptotic behavior of the signal. In massive theories ( $d=1$ ), the signal oscillates with a decaying amplitude ( $\propto t^{-d/2}$ ). In massless theories ( $d=1$ ), the signal approaches a constant non-zero value at long times, whereas in $d=3$ , it exhibits exponential decay, posing challenges for measurement.
- The "light cone" of the disturbance is shown to depend on the smearing scale $\epsilon$ ; for $\epsilon \ll M^{-1}$ , the disturbance propagates at the speed of light, while for $\epsilon \gg M^{-1}$ , the front propagates slower.
Nonlocal Quench Proposal:
- To overcome the spatial limitation of local quenches (which only access fields at a single point), the authors propose nonlocal quenches.
- By applying a perturbation involving fields at distinct points $\vec{x}_{Q,1}, \dots, \vec{x}_{Q,n}$ , the evolution of the average field at a specific point $\vec{x}_{Q,j}$ (within the time window before light cones from other points intersect) becomes proportional to the correlation function of the other $n-1$ points.
- This enables the measurement of correlations between fields localized at different spatial locations, $\langle \hat{\phi}_S(\vec{x}_{Q,1}) \dots \hat{\phi}_S(\vec{x}_{Q,n}) \rangle$ .
Analytical Solutions for Gaussian States: As a concrete demonstration, the authors apply the method to the ground state of a free massive and massless scalar field. Since the ground state is Gaussian, higher-order correlations are determined by Wick's theorem. The paper provides exact analytical formulas for the evolution of the average field in these cases, confirming that the method successfully retrieves the known 2-point correlation functions.

Significance and Claims
The paper claims to provide a new theoretical procedure for measuring correlation functions in quantum field systems, offering a pathway to study non-Gaussian correlations in initial states. The significance lies in the ability to:

Systematically access higher-order correlation functions by varying the order of the local quench operator.
Access spatially separated correlations using nonlocal quenches, a capability not available with standard local perturbations.
Distinguish between the dynamical properties of the system and the static correlations of the initial state.

The authors explicitly state that the current procedure is applicable to non-interacting theories. They note that in interacting theories, the evolution of the average field would depend on higher-order averages beyond the initial state's correlations, requiring more advanced non-equilibrium quantum field theory techniques (such as Keldysh diagram techniques) to extend the method. The paper concludes by suggesting that this approach could contribute to the field of quantum state tomography and the study of non-Gaussian correlations in many-body quantum systems, provided the method is adapted for interacting scenarios in future work.

Technical Summary: Measuring Correlations Using Local and Nonlocal Quenches

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