Quantum simulation of Abelian lattice gauge theories via state-dependent hopping

We develop a quantum simulator architecture that is suitable for the simulation of $\text{U}\left(1\right)$ Abelian gauge theories such as quantum electrodynamics. Our approach relies on the ability to control the hopping of a particle through a barrier by means of the internal quantum states of a neutral or charged impurity particle sitting at the barrier. This scheme is experimentally feasible, as the correlated hopping does not require fine-tuning of the intra- and interspecies interactions. We investigate the applicability of the scheme in a double-well potential, which is the basic building block of the simulator, both at the single-particle and the many-body mean-field level. Moreover, we evaluate its performance for different particle interactions and trapping and, specifically for atom-ion systems, in the presence of micromotion.

Figure 1

(a) Schematic illustration of the quantum simulator, where atomic particles (blue spheres) are trapped in a superlattice potential (labeled by the lattice sites $...,k-1,k,k+1,...$), whereas neutral or charged impurities (green spheres) are trapped in another periodic external potential or form an ion crystal, respectively (between the lattice sites, i.e., links). (b) Sketch of the double-well potential for the particle and the impurity with two internal states $|\uparrow \rangle$ and $|\downarrow \rangle$. (c) Sketch of the correlated hopping (left and right picture), where the impurity internal state is illustrated with a black arrow on the equatorial plane of the corresponding Bloch (green) sphere. The offset energy $\mathrm{\Delta }$ is chosen to be equal to $\hslash {\mathrm{\Omega }}_R$ (see text).

Figure 2

Sketch of the four-level particle-impurity system, where the impurity (green circle on the top of the barrier) is treated statically with two internal states, and centered in $x=0$. Here $d_L=-\sqrt{5}\ell$, $d_R=2.0\ell$, whereas $E_L^0/\left(\hslash {\omega }_L\right)=-0.5013$ and $E_R^0/\left(\hslash {\omega }_L\right)=-0.0045$ are shifted by $2\hslash {\omega }_L$ for the sake of clarity. Besides, $\mathrm{\Delta }=\delta \hslash {\omega }_L$ and $\ell =\sqrt{\hslash /\left(m_{\text{p}}{\omega }_L\right)}$. The left (blue) line and right (red) line represent the two lowest eigenfunctions of the double-well potential.

Figure 3

Panel (a): Overlap ${\mathcal{O}}_L^{+}\left(t\right)$ $[{\mathcal{O}}_R^{-}\left(t\right)$ is shown only when the coupling constants have the same strength, but opposite sign—green dashed line]. Panel (b): Correlated particle-impurity hopping $\mathcal{C}\left(t\right)$. Panel (c): Expectation value of the operator ${\widehat{G}}_L-1/2$ ($\langle {\widehat{G}}_R\rangle$ is specular to $\langle {\widehat{G}}_L\rangle$). The lines correspond to three different scenarios: $g_{1\mathrm{D}}^{e,\uparrow }=-g_{1\mathrm{D}}^{e,\downarrow }=\hslash {\omega }_L\ell$, $g_{1\mathrm{D}}^{o,\uparrow }=-g_{1\mathrm{D}}^{o,\downarrow }=0.1\hslash {\omega }_L{\ell }^3$ [solid green ${\mathcal{O}}_L^{+}\left(t\right)$ and dashed green ${\mathcal{O}}_R^{-}\left(t\right)$ line], $g_{1\mathrm{D}}^{e,\uparrow }=0.1\times g_{1\mathrm{D}}^{e,\downarrow }=\hslash {\omega }_L\ell$, $g_{1\mathrm{D}}^{o,\uparrow }=0.1\times g_{1\mathrm{D}}^{o,\downarrow }=0.1\hslash {\omega }_L{\ell }^3$ (dashed blue lines and dashed-dotted magenta lines). The only difference between the dashed blue and dashed-dotted magenta lines is how they are tuned to resonance. The dashed blue is set to have $\hslash {\mathrm{\Omega }}_R=E_R^0-E_L^0$, whereas the dashed-dotted magenta line is set to have $\hslash {\mathrm{\Omega }}_R=E_R^0-E_L^0+(J_{RR}^{\downarrow }+J_{RR}^{\uparrow }-J_{LL}^{\downarrow }-J_{LL}^{\uparrow })/2$. In all panels $m_{\text{p}}/m_{\text{i}}=1$ and ${\omega }_{\text{i}}={10}^3{\omega }_L$. Furthermore, ${\omega }_L={\omega }_R$, $d_L=\sqrt{5}\ell$, $d_R=2\ell$ for a barrier height of $4\hslash {\omega }_L$.

Figure 4

(a) Overlap ${\mathcal{O}}_L^{+}\left(t\right)$, (b) correlated particle-impurity hopping $\mathcal{C}\left(t\right)$, and (c) expectation value of ${\widehat{G}}_L-1/2$ ($\langle {\widehat{G}}_R\rangle$ is specular to $\langle {\widehat{G}}_L\rangle$) for three different scenarios: static impurity (solid cyan), moving impurity together with the quench of the barrier height (dashed red), and micromotion (dotted black). In all panels $m_{\text{p}}/m_{\text{i}}=1$, but ${\omega }_{\text{i}}=10{\omega }_L$ for the moving neutral impurity, and ${\omega }_{\text{i}}=100{\omega }_L$ for the ionic impurity micromotion with ${\mathrm{\Omega }}_{\text{rf}}/{\omega }_L=2500$. The particle-impurity interaction parameters are $g_{1\mathrm{D}}^{e,\uparrow }=-g_{1\mathrm{D}}^{e,\downarrow }=\hslash {\omega }_L\ell$, $g_{1\mathrm{D}}^{o,\uparrow }=-g_{1\mathrm{D}}^{o,\downarrow }=0.1\hslash {\omega }_L{\ell }^3$, with $\ell =\sqrt{\hslash /\left(m_{\text{p}}{\omega }_L\right)}$. Furthermore, ${\omega }_L={\omega }_R$, $d_L=\sqrt{5}\ell$, $d_R=2\ell$ for a barrier height of $4\hslash {\omega }_L$, whereas in the quenched case ${\omega }_R=1.4{\omega }_L$, $d_L\simeq 2.97\ell$, $d_R=2\ell$ for a barrier height of $5.92\hslash {\omega }_L$.

Figure 5

(a) Minimal instance of six sites and six impurities, where a demonstration of a lattice gauge simulator could be envisioned. An example of a quenched experiment is explained in the main text. (b) Gauge-invariant spectrum at half filling of the proposed model as a function of the mass parameter in the Hamiltonian (16). As it can be seen from the degeneracy of the low-energy spectrum, there is a region with a unique ground state and a region with a twofold degenerate ground-state subspace. The numerical calculation has been performed by exact diagonalization and by assuming periodic boundary conditions.

Figure 6

(a) Probabilities of the three states that characterize the different phases of the model: (dashed green) probability of being in a disordered state; (solid blue and dotted red) probability of being in one of the two charge and parity broken states, where the system is prepared in the blue state. (b) Evolution in real time of the electric flux density in a quenched experiment, where the system is initialized in the same state as in (a), with a nonzero value of the electric flux and it evolves oscillating around a zero value.

Figure 7

(a) Eigenfunctions of the harmonic potential, ${\psi }_n\left(x\right)$, for $d_R=0$, $\delta =0$, and $r=1$. (b) Eigenfunctions for the symmetric double harmonic potential $d_R=2$, $\delta =0$, and $r=1$. (c) Eigenfunctions for the tilted double harmonic potential for $d_R=2$, $\delta =2$, and $r=1$. (d) Eigenfunctions for a narrower tilted double harmonic potential, namely for $d_R=2$, $\delta =2$, and $r=1.4$. For the sake of better visibility, all eigenfunctions have been located on the vertical (energy) axis separated by one $\hslash {\omega }_L$. Besides, in all panels $d_R$ and $\delta$ are in units of $\ell$ and $\hslash {\omega }_L$, respectively.

Figure 8

Green solid $\left[{\mathcal{O}}_L\left(t\right)\right]$ and dotted $\left[{\mathcal{O}}_R\left(t\right)\right]$ lines show the result of the correlated hopping with a condensate by means of the compensating control pulse technique (D2), while blue dashed line shows the results from machine learning techniques. The simulation corresponds to $N=10$, ${}^{87}\mathrm{Rb}$ bosonic atoms with an interaction strength $g=0.21\hslash {\omega }_L\ell$ that assumes ${\omega }_L=2\pi 1$ kHz and a transverse trap frequency $10{\omega }_L$. Panel (a) shows the overlap fidelity, panel (b) the correlation function, and panel (c) the expectation value of the operator ${\widehat{G}}_L$.