Simulating Compact Quantum Electrodynamics with Ultracold Atoms: Probing Confinement and Nonperturbative Effects

Recently, there has been much interest in simulating quantum field theory effects of matter and gauge fields. In a recent work, 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 us 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.

Figure 1

Single plaquette plots. (a) Graphs of $⟨E_1(g)⟩$. Black—the calculation of Ref. 5—small coupling approximation for regular Abelian Kogut-Susskind theory. Blue—exact calculation for the truncated theory (tKS), for $l=1,\dots ,20$. Red—exact calculation for the spin-gauge theory (SG), for $l=1,\dots ,20$. It can be seen that the curves start to merge for a small $g$ and $l$. The value 1 refers to the flux tube and $\frac{3}{4}$ to the longitudinal part. (b) Graphs of $P_l(g)$, for $l=1,\dots ,20$. It can be seen that even for small values of $l$, $P_l(g)$ approaches 1 for a small $g$. (c) The one plaquette system we use in the demonstration.

Figure 2

(a) The simulation lattice. As explained in the text, the atoms (yellow circles) are aligned along the links of a 2D square lattice, with basis vectors $\widehat{\mathbf{1}}$ and $\widehat{\mathbf{2}}$. On the blue (horizontal/vertical) lines the tunneling rates between neighboring atoms are $t_{\alpha }^s$ [in Hamiltonian (2)], and on the red (diagonal) lines—$t_{\alpha }^d$. In Hamiltonian (3), $z-z$ interaction is between nearest neighbors along both the red and blue lines, and $x-x$, $y-y$ interactions are only along the red ones. (b) The flux tube (zeroth order in perturbative expansion, in the strong limit of the Kogut-Susskind Hamiltonian) connecting two opposite charges. (c) The flux tube in the language of our simulating system.

Figure 3

An example of the emergence of gauge invariance in the second effective calculations for a single plaquette system ($l=1$). Two unit charges are located in the lower vertices. The parameters are chosen such that for a large $\lambda$ the system will be in the strong limit: $\mu =1$, $\Omega =0.1$. The expectation values of $L_z$ along two links, as well as of the gauge transformation generators $G$ at two vertices are plotted as a function of $\lambda$ for the ground state. It can be seen that as $\lambda$ grows, $G$ converges to the local charges and hence Gauss’s law and gauge invariance are introduced, and $L_z$ approach the values of the lowest flux-tube state in the strong limit. Note the discussion on the signs in the text.