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Simulating Compact Quantum Electrodynamics with ultracold atoms: Probing confinement and nonperturbative effects
Erez Zohar, J. Ignacio Cirac, Benni Reznik
TL;DR
Simulating cQED is motivated by its confinement behavior and nonperturbative effects. This paper proposes a single-atom spin-gauge model whose convergence to cQED improves with the number of internal levels. The authors report convergence in weak and strong coupling and give an l=1 construction for confinement between external static charges.
Problem
cQED exhibits confinement related to nonperturbative effects, and lattice gauge theories are believed to have nontrivial phase structure.
Source: p. 1Method
The authors propose a spin-gauge Hamiltonian using single optical-lattice atoms with 2l+1 internal levels to simulate cQED.
Source: p. 1Results
The spin-gauge Hamiltonian rapidly converges to the Kogut-Susskind model in both weak and strong coupling; for g < 1, l≈2–3 gives agreement, while at large g the flux-tube value of 1 is reached at l=1.
Source: pp. 1–2Takeaways & Limitations
The rapid convergence suggests a route to simulating weak-coupling nonperturbative effects and phase transitions in 3+1 dimensions.
Source: p. 5Predictions for the proposed measurements are based on perturbation theory, so realizing them would also test those predictions.
Source: State preparation and possible… · p. 6
Abstract
from arXiv · showhide
Recently, there has been much interest in simulating quantum field theory effects of matter and gauge fields. In a recent work [Phys. Rev. Lett. 107, 275301 (2011)] a method for simulating compact Quantum Electrodynamics (cQED) using Bose-Einstein condensates has been suggested. We suggest an alternative approach, which relies on single atoms in an optical lattice, carrying 2l+1 internal levels, which converges rapidly to cQED as l increases. That enables the simulation of cQED in 2+1 dimensions in both the weak and the strong coupling regimes, hence allowing to probe confinement as well as other nonperturbative effects of the theory. We provide an explicit construction for the case l=1 which is sufficient for simulating the effect of confinement between two external static charges.
The superlattice structure
The proposed l=1 simulation uses a two-dimensional optical superlattice to implement the spin-gauge Hamiltonian for cQED. Occupied sites sit on model links, while auxiliary sites and laser-assisted tunneling generate the required hopping processes.
The superlattice structure
- The l=1 spin-gauge Hamiltonian is proposed as an optical-lattice simulation model for cQED.The construction uses single atoms with three internal levels on a two-dimensional square lattice.Source: The superlattice structure · p. 5
- A superlattice with unoccupied auxiliary minima uses laser coupling and adiabatic elimination to generate hopping between occupied sites.The two auxiliary-site types have different depths.Source: The superlattice structure · p. 5
- Occupied yellow minima sit on the model links, while blue and red auxiliary sites of different depths mediate tunneling between them.The lattice is shown rotated so model links appear as vertices.Source: Figure 4 · p. 5, The superlattice structure · p. 5
- Lasers couple m=±1 states through blue auxiliary sites and m=0 states through red sites, generating the required tunneling rates.No additional lasers are needed because t_s^0 is set to zero.Source: The superlattice structure · p. 5
State preparation and possible measurements
The preparation scheme starts from the strong-coupling gauge-field vacuum and can add plaquette loops or manually prepared flux excitations. Freezing measured species densities provides a qualitative glimpse of finite loops and charge-connected flux tubes, with predictions based on perturbation theory.
State preparation and possible measurements
- With bosonic interactions off, the strong-coupling gauge-field vacuum has every atom in m=0 and no excited links.This is identified as the exact QED vacuum in the extreme strong limit.Source: State preparation and possible… · p. 5
- The system must first be prepared in an eigenstate of H_G + H_E; the constraint and electric-field terms need not be turned on adiabatically.Plaquette terms are absent in this initial state.Source: State preparation and possible… · p. 6
- Adiabatically increasing Ω can dress the state into a first-order QED vacuum or a loop sea, depending on the chosen ramp regime.A loop sea is prepared at Ω²/λ ≫ μ; Ω is later lowered for strong-regime QED dynamics.Source: State preparation and possible… · p. 6
- Single-addressing lasers can manually create closed flux loops or alternating-sign flux tubes before plaquette interactions are ramped on.This route is accurate only to zeroth order, though corrections appear.Source: State preparation and possible… · p. 6
- Freezing species densities after measurement collapses the state to one superposition element, revealing finite loops or flux tubes between static charges.The prepared zeroth-order flux tube should remain, potentially with an additional loop elsewhere on the lattice.Source: State preparation and possible… · p. 6
- The proposed measurement predictions are based on perturbation theory, making their realization an experimental check of those predictions.Source: State preparation and possible… · p. 6
Accuracy of Hl eigenstates in perturbation theory
The perturbative accuracy of the truncated Spin-Gauge Hamiltonian is controlled by the cutoff l, the unperturbed eigenvalue bound n0, and the maximum ladder-operator power m. For compact lattice QED, this makes the required cutoff higher when external charges are present.
Accuracy of Hl eigenstates in perturbation theory
- Nth-order accuracy requires l ≥ n0 + nm, so the maximum accurate order is n ≤ (l − n0)/m.Here m is the maximal power of a raising or lowering operator in the perturbation.Source: Accuracy of Hl eigenstates in… · pp. 6–7
- For compact QED, m = 1: l = 1 gives first-order accuracy without external charges but only zeroth-order accuracy with charges; l = 2 is needed for first order with charges.The two-charge case has n0 = 1, reducing the maximum accurate order to n ≤ l − 1.Source: Accuracy of Hl eigenstates in… · p. 7
- For l = 1, the ladder operators are nonunitary, but their nonzero matrix elements match; local counterterms provide a cure for this issue at l = 2.The authors therefore disregard the nonunitarity issue in the l = 1 case.Source: Accuracy of Hl eigenstates in… · p. 7