Digital lattice gauge theories

We propose a general scheme for a digital construction of lattice gauge theories with dynamical fermions. In this method, the four-body interactions arising in models with $2+1$ dimensions and higher are obtained stroboscopically, through a sequence of two-body interactions with ancillary degrees of freedom. This yields stronger interactions than the ones obtained through perturbative methods, as typically done in previous proposals, and removes an important bottleneck in the road towards experimental realizations. The scheme applies to generic gauge theories with Lie or finite symmetry groups, both Abelian and non-Abelian. As a concrete example, we present the construction of a digital quantum simulator for a ${\mathbb{Z}}_3$ lattice gauge theory with dynamical fermionic matter in $2+1$ dimensions, using ultracold atoms in optical lattices, involving three atomic species, representing the matter, gauge, and auxiliary degrees of freedom, that are separated in three different layers. By moving the ancilla atoms with a proper sequence of steps, we show how we can obtain the desired evolution in a clean, controlled way.

Figure 1

A single plaquette. It is labeled after the vertex on the left bottom, $\mathbf{x}$, where the fermionic spinor ${\psi }_m^{j†}\left(\mathbf{x}\right)$ resides. The gauge-field operators $U_{mn}^j\left(\mathbf{x},1\right)$ and $U_{mn}^j\left(\mathbf{x},2\right)$ act on the gauge-field Hilbert spaces of the links $\left(\mathbf{x},1\right)$ and $\left(\mathbf{x},2\right)$ respectively.

Figure 2

The ${\mathbb{Z}}_3$ simulating system consists of three layers, to avoid undesirable interactions. The lowest layer (red) is for the fermionic matter at the vertices, the highest one (blue) is for the link atoms (simulating the gauge field), and the ancilla atoms (green) are trapped “at rest” in an intermediate layer, from which they can rigidly move to interact with the other atoms, above or below them (see green lines). Full circles denote minima which are initially occupied.

Figure 3

The ${\mathbb{Z}}_3$ simulating system consists of three atomic species, trapped in three different layers. The simulated system is planar, and contains gauge fields on the links (blue circles, with three possible atomic levels), matter fermions on the vertices (represented by red circles) and control, ancilla atoms (green) located at the centers of every second plaquette, as described in the text, with three internal levels as well.

Figure 4

The complete ${\mathbb{Z}}_3$ sequence, for a single time step $W$ (simulating time $\tau$). (1) Two plaquettes—the left is even, the right is odd. Blue circles denote the gauge field atoms, red ones the matter fermions (which can be present or absent) and green ones the control atoms, initially in the centers of the even plaquettes. (2) ${\mathcal{U}}_{ev}$, creating a stator for the even vertical links [here and in the following, blue lines (squares) show that a stator is active for the corresponding links (plaquettes)]. (3) The controls interact with fermions to realize ${\mathcal{U}}_W^{ev†}$. (4) Fermionic tunneling for even vertical links is allowed: ${\mathcal{U}}_t$ (here and in the following denoted by a red line connecting a pair of fermions). (5) ${\mathcal{U}}_W^{ev}$. (6) The stator is undone. (7)–(10) A similar process for even horizontal links, but without undoing the stators. (11)–(13) Plaquette stators are completed for even plaquettes. (14) ${\tilde{V}}_B$ is generated by addressing the control atoms (local operations on atoms are denoted by red circles). This will eventually give rise to $W_{Be}$. (15)–(17) The plaquettes' stators are undone, but stators for odd vertical links are kept. (18) The controls' “rest positions” are moved to the odd plaquettes on the right. (19)–(34) Similar process is repeated for the odd plaquettes, but the stators eventually are completely undone in this case, for the next step. (35) The controls are moved back to the centers of the even plaquettes, and the link and vertex atoms are evolved according to the noninteracting parts, $W_E$ and $W_M$ (again noninteracting terms are denoted by red circles). Then the whole sequence can be repeated.

Figure 5

The external, static vector potential for each link, which results from the physical realization, for both an even plaquette (left) and an odd one (right). The lattice curl for both types of plaquette is zero, which means that these phases do not influence the physics and may be removed or, practically, be ignored.