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Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator

Bing Yang, Hui Kai Sun, R.J. Ott, Han-Yi Wang, Torsten V. Zache, Jad C. Halimeh, Zhen-Sheng Yuan, Philipp Hauke, Jian-Wei Pan

Published Jan 1, 2020• 273 citations

Abstract

Abstract The modern description of elementary particles, as formulated in the standard model of particle physics, is built on gauge theories1. Gauge theories implement fundamental laws of physics by local symmetry constraints. For example, in quantum electrodynamics Gauss’s law introduces an intrinsic local relation between charged matter and electromagnetic fields, which protects many salient physical properties, including massless photons and a long-ranged Coulomb law. Solving gauge theories using classical computers is an extremely arduous task2, which has stimulated an effort to simulate gauge-theory dynamics in microscopically engineered quantum devices3–6. Previous achievements implemented density-dependent Peierls phases without defining a local symmetry7, 8, realized mappings onto effective models to integrate out either matter or electric fields9–12, or were limited to very small systems13–16. However, the essential gauge symmetry has not been observed experimentally. Here we report the quantum simulation of an extended U(1) lattice gauge theory, and experimentally quantify the gauge invariance in a many-body system comprising matter and gauge fields. These fields are realized in defect-free arrays of bosonic atoms in an optical superlattice of 71 sites. We demonstrate full tunability of the model parameters and benchmark the matter–gauge interactions by sweeping across a quantum phase transition. Using high-fidelity manipulation techniques, we measure the degree to which Gauss’s law is violated by extracting probabilities of locally gauge-invariant states from correlated atom occupations. Our work provides a way to explore gauge symmetry in the interplay of fundamental particles using controllable large-scale quantum simulators.

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Can high-energy physics be simulated by low-energy, non-relativistic, many-body systems such as ultracold atoms? Such ultracold atomic systems lack the type of symmetries and dynamical properties of high energy physics models: in particular, they manifest neither local gauge invariance nor Lorentz invariance, which are crucial properties of the quantum field theories which are the building blocks of the standard model of elementary particles. However, it turns out, surprisingly, that there are ways to configure an atomic system to manifest both local gauge invariance and Lorentz invariance. In particular, local gauge invariance can arise either as an effective low-energy symmetry, or as an exact symmetry, following from the conservation laws in atomic interactions. Hence, one could hope that such quantum simulators may lead to a new type of (table-top) experiments which will be used to study various QCD (quantum chromodynamics) phenomena, such as the confinement of dynamical quarks, phase transitions and other effects, which are inaccessible using the currently known computational methods. In this report, we review the Hamiltonian formulation of lattice gauge theories, and then describe our recent progress in constructing the quantum simulation of Abelian and non-Abelian lattice gauge theories in 1 + 1 and 2 + 1 dimensions using ultracold atoms in optical lattices.

arXiv:1503.02312•Reports on Progress in Physics•Dec 18, 2015•426 citations

Abstract The modern description of elementary particles, as formulated in the standard model of particle physics, is built on gauge theories1. Gauge theories implement fundamental laws of physics by local symmetry constraints. For example, in quantum electrodynamics Gauss’s law introduces an intrinsic local relation between charged matter and electromagnetic fields, which protects many salient physical properties, including massless photons and a long-ranged Coulomb law. Solving gauge theories using classical computers is an extremely arduous task2, which has stimulated an effort to simulate gauge-theory dynamics in microscopically engineered quantum devices3–6. Previous achievements implemented density-dependent Peierls phases without defining a local symmetry7, 8, realized mappings onto effective models to integrate out either matter or electric fields9–12, or were limited to very small systems13–16. However, the essential gauge symmetry has not been observed experimentally. Here we report the quantum simulation of an extended U(1) lattice gauge theory, and experimentally quantify the gauge invariance in a many-body system comprising matter and gauge fields. These fields are realized in defect-free arrays of bosonic atoms in an optical superlattice of 71 sites. We demonstrate full tunability of the model parameters and benchmark the matter–gauge interactions by sweeping across a quantum phase transition. Using high-fidelity manipulation techniques, we measure the degree to which Gauss’s law is violated by extracting probabilities of locally gauge-invariant states from correlated atom occupations. Our work provides a way to explore gauge symmetry in the interplay of fundamental particles using controllable large-scale quantum simulators.

Quantum simulations of high-energy physics, and especially of gauge theories, is an emerging and exciting direction in quantum simulations. However, simulations of such theories, compared to simulations of condensed matter physics, must satisfy extra restrictions, such as local gauge invariance and relativistic structure. In this paper we discuss these special requirements, and present a method for quantum simulation of lattice gauge theories using ultracold atoms. This method allows us to include local gauge invariance as a fundamental symmetry of the atomic Hamiltonian, arising from natural atomic interactions and conservation laws (and not as a property of a low-energy sector). This allows us to implement elementary gauge invariant interactions for three lattice gauge theories: U(1) (compact QED),

{\mathbb{Z}}_{N}

and SU

(N)

(Yang-Mills), which can be used to build quantum simulators in

1+1

dimensions. We also present a loop method, which uses the elementary interactions as building blocks in the effective construction of quantum simulations for

d+1

dimensional lattice gauge theories (

d>1

), but unlike in previous proposals, here gauge invariance and Gauss's law are natural symmetries, which do not have to be imposed as a constraint. We discuss in detail the quantum simulation of

2+1

dimensional compact QED and provide a numerical proof of principle. The simplicity of the already gauge-invariant elementary interactions of this model suggests it may be useful for future experimental realizations.

Abstract A major driver of quantum-simulator technology is the prospect of probing high-energy phenomena in synthetic quantum matter setups at a high level of control and tunability. Here, we propose an experimentally feasible realization of a large-scale 2 + 1D U(1) gauge theory with dynamical matter and gauge fields in a cold-atom quantum simulator with spinless bosons. We present the full mapping of the corresponding Gauss’s law onto the bosonic computational basis. We then show that the target gauge theory can be faithfully realized and stabilized by an emergent gauge protection term in a two-dimensional single-species Bose–Hubbard optical Lieb superlattice with two spatial periods along either direction, thereby requiring only moderate experimental resources already available in current cold-atom setups. Using infinite matrix product states, we calculate numerical benchmarks for adiabatic sweeps and global quench dynamics that further confirm the fidelity of the mapping. Our work brings quantum simulators of gauge theories a significant step forward in terms of investigating particle physics in higher spatial dimensions, and is readily implementable in existing cold-atom platforms.

In the fundamental laws of physics, gauge fields mediate the interaction between charged particles. An example is the quantum theory of electrons interacting with the electromagnetic field, based on U(1) gauge symmetry. Solving such gauge theories is in general a hard problem for classical computational techniques. Although quantum computers suggest a way forward, large-scale digital quantum devices for complex simulations are difficult to build. We propose a scalable analog quantum simulator of a U(1) gauge theory in one spatial dimension. Using interspecies spin-changing collisions in an atomic mixture, we achieve gauge-invariant interactions between matter and gauge fields with spin- and species-independent trapping potentials. We experimentally realize the elementary building block as a key step toward a platform for quantum simulations of continuous gauge theories.

Despite the importance of gauge theories for modern physics, solving their out-of-equilibrium dynamics on classical computers is extremely challenging. This difficulty is currently stimulating a worldwide effort to implement them in dedicated quantum simulators. In this talk, I will discuss recent progress towards quantum simulation of gauge theories using ultracold atoms. I will show recent breakthroughs to overcome the main challenge, the realization of a dynamics that respects gauge symmetry. In particular, I will present a recent experiment based on engineering of suitable energy penalties in an optical lattice, which has realized a many-body gauge theory in a 71-site Hubbard model and has certified the fulfilment of Gauss’s law for the first time. Further, I will discuss recent progress in understanding gauge breaking errors in this and related models as well as possibilities to mitigate them. I will also illustrate some of the fascinating physics attainable in quantum simulators of gauge theories even for simple target models and with realistic resources, such as dynamical topological quantum phase transitions. Finally, I will discuss the state of art and common issues of gauge quantum simulation, and thus aim at outlining a roadmap towards mature and practically relevant quantum simulation of gauge theories.

Large-Scale

2+1

D

\mathrm{U}(1)

Gauge Theory with Dynamical Matter in a Cold-Atom Quantum Simulator

A major driver of quantum-simulator technology is the prospect of probing high-energy phenomena in synthetic quantum matter setups at a high level of control and tunability. Here, we propose an experimentally feasible realization of a large-scale

2+1

D

\mathrm{U}(1)

gauge theory with dynamical matter and gauge fields in a cold-atom quantum simulator with spinless bosons. We present the full mapping of the corresponding Gauss's law onto the bosonic computational basis. We then show that the target gauge theory can be faithfully realized and stabilized by an emergent gauge protection term in a two-dimensional single-species Bose--Hubbard optical Lieb superlattice with two spatial periods along either direction, thereby requiring only moderate experimental resources already available in current cold-atom setups. Using infinite matrix product states, we calculate numerical benchmarks for adiabatic sweeps and global quench dynamics that further confirm the fidelity of the mapping. Our work brings quantum simulators of gauge theories a significant step forward in terms of investigating particle physics in higher spatial dimensions, and is readily implementable in existing cold-atom platforms.

Can high-energy physics be simulated by low-energy, non-relativistic, many-body systems such as ultracold atoms? Such ultracold atomic systems lack the type of symmetries and dynamical properties of high energy physics models: in particular, they manifest neither local gauge invariance nor Lorentz invariance, which are crucial properties of the quantum field theories which are the building blocks of the standard model of elementary particles. However, it turns out, surprisingly, that there are ways to configure an atomic system to manifest both local gauge invariance and Lorentz invariance. In particular, local gauge invariance can arise either as an effective low-energy symmetry, or as an exact symmetry, following from the conservation laws in atomic interactions. Hence, one could hope that such quantum simulators may lead to a new type of (table-top) experiments which will be used to study various QCD (quantum chromodynamics) phenomena, such as the confinement of dynamical quarks, phase transitions and other effects, which are inaccessible using the currently known computational methods. In this report, we review the Hamiltonian formulation of lattice gauge theories, and then describe our recent progress in constructing the quantum simulation of Abelian and non-Abelian lattice gauge theories in 1 + 1 and 2 + 1 dimensions using ultracold atoms in optical lattices.