Quantum Sensors for the Generating Functional of Interacting Quantum Field Theories

Difficult problems described in terms of interacting quantum fields evolving in real time or out of equilibrium abound in condensed-matter and high-energy physics. Addressing such problems via controlled experiments in atomic, molecular, and optical physics would be a breakthrough in the field of quantum simulations. In this work, we present a quantum-sensing protocol to measure the generating functional of an interacting quantum field theory and, with it, all the relevant information about its in- or out-of-equilibrium phenomena. Our protocol can be understood as a collective interferometric scheme based on a generalization of the notion of Schwinger sources in quantum field theories, which make it possible to probe the generating functional. We show that our scheme can be realized in crystals of trapped ions acting as analog quantum simulators of self-interacting scalar quantum field theories.

Popular Summary

Quantum field theory (QFT) is the central theoretical framework for understanding subatomic particles and their interactions. The theory provides a unifying language for a wide variety of systems across many energy scales, from ultracold atoms in the laboratory to ultrarelativistic particles at the Large Hadron Collider. A cornerstone of QFT is the generating functional, a mathematical tool that neatly compresses all the relevant information about the QFT into a single, somewhat abstract, quantity. From the generating functional, all correlation functions in QFT can be derived. Usually thought of as purely mathematical, we show how the generating functional can, in fact, be measured in the lab, using experiments in atomic, molecular, and optical physics. This would allow one to answer a wide range of complicated questions about nature.

Our detailed protocol shows how to map the generating functional onto a collection of entangled quantum sensors and how to recover the mapped functional using interferometric experiments. We describe how to implement this protocol using systems of trapped atomic ions for a self-interacting quantum scalar field, a paradigm QFT for symmetry breaking with applications to Bose condensation and Higgs physics, and we discuss possible generalizations to other QFTs.

This result constitutes an important step in the broader topic of quantum simulations, which aim to understand problems in quantum many-body physics by means of experimental systems that can be accurately manipulated to represent the QFT under investigation.

Figure 1

Schematic representation of the Schwinger sensors for the generating functional. We represent a quantum scalar field $\varphi (x)$ in a $D=2+1$ space-time, which is discretized into a $d=2$ spatial lattice, while letting the time coordinate continuous. Inset: Zoom of a small space region, where the field at each point (red circles) is coupled to the fields at neighboring points (small springs) and can be excited by its coupling (wavy lines) to generalized quantum-mechanical Schwinger sources (green circles with arrows). These ${\mathbb{Z}}_2$ Schwinger sources will also function as quantum sensors for the generating functional.

Figure 2

Schematic representation of the quantum sensing for Feynman propagators. The different indexes for the lattice sites $\mathbf{x}\in {\mathrm{\Lambda }}_s$, as well as the corresponding ${\mathbb{Z}}_2$ sensors labeled by ${\mathbf{x}}^s$, are mapped onto the keys of a piano. The set of pulse sequences ${\mathbf{J}}^{(m)}$ that couple the sensors to the field Eq. (14) corresponds to a piano score that indicates the sequence of keys (sensors) that must be pressed (coupled to the field) at different instants of time to produce a melody (spin-parity measurement) that encodes the relevant information about the Feynman propagators.

Figure 3

Effective $\lambda {\varphi }^4$ QFT for ion crystals and quantum sensing scheme. Left: In the vicinity of the linear-to-zigzag structural phase transition of a trapped-ion crystal, the transverse zigzag vibrations yield the soft mode that contains the universal properties of the transition. Right: The slowly varying envelope of the zigzag distortion Eq. (32) allows us to develop a gradient expansion that leads to the effective $\lambda {\varphi }^4$ QFT. By exploiting two electronic levels of the ions, and a state-dependent dipole force Eq. (49), one can infer the generating functional of the QFT.