Ab Initio Derivation of Lattice-Gauge-Theory Dynamics for Cold Gases in Optical Lattices

We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent optical lattices. The proposal enables the study of both gauge and fermionic matter fields without integrating out one of them in one and two dimensions. We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species. Starting from an ab initio modeling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory. This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theories that—while pivotal for a successful implementation—have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics. We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope ${}^{173}$$\mathrm{Yb}$, addressing via simulations all key sources of imperfections, and provide concrete parameter estimates for relevant energy scales in both one- and two-dimensional settings.

Popular Summary

The simulation of strongly interacting quantum many-body systems is one of the most natural applications of quantum technology and one of the most promising for practical quantum advantage. At the interface between particle physics and quantum information, a very active line of research has been quantum simulation of gauge theories. Gauge theories play a fundamental role in many research areas, ranging from condensed-matter to high-energy physics. Recent experimental realizations have demonstrated first proof-of-principle realizations. However, making significant advances in the field will require the engineering of more sophisticated optical potentials and will rely on the manipulation of many degrees of freedom. Due to the increased complexity, it is crucial to accurately determine the energy scales of the relevant physics and to ensure that noise and decoherence effects are insignificant over sufficiently long experimental time scales.

In this work, we develop a novel scheme for the quantum simulation of a U(1) lattice gauge theory in one and two spatial dimensions with fermionic alkaline-earth atoms in optical lattices and present a detailed ab initio derivation of the associated energy scales. Through comparison with known experimental limitations, we are able to demonstrate the practical realizability of the proposed scheme. Our results thus indicate a direct way for quantum simulation of a lattice gauge theory beyond one spatial dimension, an accomplishment that has not been achieved to date.

Figure 1

The mapping between atomic states in the optical-lattice potential and the U(1) quantum link model considered in our proposal. (a) A schematic of the optical-lattice potentials for the $g$ (blue line) and $e$ atoms (orange line), together with the relevant energy scales ${\delta }_{g,e}$ and ${\mathrm{\Delta }}_{g,e}$. The circles indicate an exemplary initial state of $g$ (blue circles) and $e$ atoms (orange circles) in the optical lattice. The black arrows on the right indicate the correlated hopping of atoms between adjacent lattice sites. (b) A simplified schematic of the optical-lattice potential in (a) showing the bonds across which hopping is energetically allowed (thick gray lines) and forbidden (dashed gray lines). (c) The state in the quantum link model corresponding to the atomic configuration in (a) and (b). In the mapping, every lattice site that can only be occupied by $e$ or $g$ atoms is interpreted as a matter site, shown as circles, with the plus ($+$) [minus ($-$)] labels indicating filled sites with positive ($+$) [negative ($-$)] charge. The lattice sites that can be occupied by both $e$ and $g$ are interpreted as link $(l)$ or gauge-field sites, where the triangles pointing right (left) indicate a positive (negative) electric field. The colors indicate the corresponding internal state of the atoms [see (a) and (b)] in the proposed experimental implementation and gray indicates an empty site. Note that the correlated hopping of atoms is mapped to a gauge-invariant pair creation-annihilation process, as shown on the right.

Figure 2

The optical-lattice potential and the Wannier functions. The $x$ component of the optical-lattice potential defined in Eq. (6) for the (a) $g$ and (b) $e$ atoms is plotted in gray for $\phi =0$. The parameters $A_{\alpha }$, $B_{\alpha }$, and $C_{\alpha }$ are reported in Table 1. The $x$ components of the Wannier functions centered on sites $x=3a$ and $x=2a$ are shown in red. The parameter $s=\{-,0,+\}$ labels the three different orbitals in the unit cell.

Figure 3

The interacting terms in $H_U$ and $H_D$. (a) The on-site interaction $U$ between a single $g$ (blue circle) and $e$ atom (orange circle). (b) The hopping of a single $e$ atom to an empty lattice site. (c) In the presence of interactions, tunneling is additionally modified by a density-assisted tunneling with amplitude $D_e$.

Figure 4

Examples of the mapping between atomic configurations in the optical lattice and gauge-invariant states in the U(1) QLM. In each panel, the atomic states in the optical lattice are shown in the top row and the states in the QLM are shown in the bottom row. (a) The state with neighboring pairs of $+$ and $-$ charges. (b) The vacuum state with a homogeneous negative electric field $E=-1/2$. (c) The vacuum state with a homogeneous positive electric field $E=+1/2$.

Figure 5

The dynamics in the $\text{U(1)}$ quantum link model. The time evolution of (a) the number of atoms per link and (b) [(c)] the number of atoms on each odd (even) block. For each observable $O$, the shaded area indicates the interval $⟨O⟩\pm \sqrt{\mathrm{Var}(O)}$. The initial state is depicted in Fig. 4 and is evolved under $H_{\mathrm{latt}}$. The system has periodic boundary conditions and finite size $4a$. The top-left schematic in each panel illustrates the observable. (d) The time evolution of the electric field on an odd-even link. The exact dynamics given by $H_{\mathrm{latt}}$ are compared with the second-order effective Hamiltonian [using Eqs. (18) and (19)] and with the fourth-order effective Hamiltonian $H_{\mathrm{eff}}^{(4)}$. The latter two are both equivalent to $H_{\mathrm{QLM}}$. (e) The difference $\mathrm{\Delta }{E}_{\text{odd,even}}=⟨E_{\text{odd,even}}{⟩}_{H_{\mathrm{latt}}}-⟨E_{\text{odd,even}}{⟩}_H$ (the subscript denotes the Hamiltonian that generates the time evolution) for the two cases $H=H_{\mathrm{QLM}}$ and $H=H_{\mathrm{eff}}^{(4)}$.

Figure 6

The Fourier transform of the time evolution of the electric field. (a) The spectrum corresponding to the time evolution shown in Fig. 5. (b) An enlargement of (a), revealing details at small frequencies.

Figure 7

The influence of disorder on the dynamics of the electric field. The Fourier transform of the time evolution of the electric field under $H_W$ for different values of $W$. The legend indicates the values of $W/h$ in kilohertz. The results are averaged over 100 disorder realizations.

Figure 8

The implementation of the quantum link model in two dimensions. (a),(b) The optical lattice for the (a) $g$ and (b) $e$ atoms. (c) An example of a gauge-invariant state belonging to the resonant subspace. The blue and orange circles represent $g$ and $e$ atoms, respectively. (d) The corresponding gauge-invariant state in the QLM. The dark- (light-) gray circles indicate the occupied (empty) matter sites (with charge $+$, $-$, or no charge). The red (blue) arrows represent a link with electric field $E_{r,r+k}=+1/2$ ($E_{r,r+k}=-1/2$).

Figure 9

The lattice potential and band structure along $x$. (a) The $x$ component of the optical-lattice potential $V_g^x(x)$. The defining parameters are reported in Table 1. (b) The corresponding band structure, using the same scale as in (a). The two lowest bands are almost degenerate (they correspond to symmetric and antisymmetric superpositions of the $s=\{+,-\}$ sites in the triple well) and are separated from the third one (corresponding to the central site $0$ of the triple well) by an energy of approximately ${\delta }_g$. Higher bands are separated from the first three by an energy gap ${\mathrm{\Delta }}_g$.

Figure 10

The band structure for the 2D lattice. The ten lowest bands corresponding to the 2D lattice $V_e^{x,y}(x,y)$ in Fig. 8.

Figure 11

The Wannier functions for the 2D lattice. The 2D Wannier functions of two $g$ and two $e$ blocks. In orange (light blue), we plot the four Wannier functions localized on the links of each block for the $e$ ($g$) state. In red (dark blue), we plot the Wannier function localized on the center of each block for the $e$ ($g$) state.