Quantum Simulation of a Lattice Schwinger Model in a Chain of Trapped Ions

We discuss how a lattice Schwinger model can be realized in a linear ion trap, allowing a detailed study of the physics of Abelian lattice gauge theories related to one-dimensional quantum electrodynamics. Relying on the rich quantum-simulation toolbox available in state-of-the-art trapped-ion experiments, we show how one can engineer an effectively gauge-invariant dynamics by imposing energetic constraints, provided by strong Ising-like interactions. Applying exact diagonalization to ground-state and time-dependent properties, we study the underlying microscopic model and discuss undesired interaction terms and other imperfections. As our analysis shows, the proposed scheme allows for the observation in realistic setups of spontaneous parity- and charge-symmetry breaking, as well as false-vacuum decay. Besides an implementation aimed at larger ion chains, we also discuss a minimal setting, consisting of only four ions in a simpler experimental setup, which enables us to probe basic physical phenomena related to the full many-body problem. The proposal opens a new route for analog quantum simulation of high-energy and condensed-matter models where gauge symmetries play a prominent role.

Popular Summary

Gauge invariance appears as a fundamental symmetry in nature, common to electromagnetism and the standard model of particle physics, and it can emerge in effective theories in solid-state physics. At a microscopic level, gauge invariance governs the interactions between subatomic particles, giving rise to complex quantum field theories. In principle, these gauge theories allow us to gain precise understanding of the microscopic interactions of the system. In practice, however, calculations of physical quantities based on gauge theories often exceed current computational capabilities. These difficulties could be overcome by quantum simulators, inspired by Feynman’s original idea of employing a quantum machine to tackle a fully quantum-mechanical problem. Motivated by the very rapid experimental developments in controlling systems of cold trapped ions, we show concretely in this paper that one of the simplest instances of quantum gauge theories, the one-dimensional version of quantum electrodynamics, can be realized in state-of-the-art ion setups.

Within the proposed experimental setup, the artificial electron and photon degrees of freedom of the target gauge theory are encoded in the internal state of the ions. Laser light permits us to precisely control these degrees of freedom, allowing us to initialize the system and engineer gauge-invariant spin interactions between the ions. Furthermore, here we show how gauge invariance can be imposed by shaping the interactions between the internal states such that the Gauss law of electrodynamics constrains the system dynamics.

Our work paves the way towards a controlled realization of gauge models in systems of trapped ions, with the potential of providing insights into the in- and out-of-equilibrium physics of complex many-body problems present in high-energy and condensed-matter physics.

Figure 1

(a) Lattice gauge theory. Fermions with annihilation operators ${\psi }_i$, living on lattice sites $i$, couple to gauge fields, represented by the operators $U_{i,i+1}$, living on links $i$, $i+1$. (b) A fermion tunnels from site $i$ to $i-1$ and flips the gauge field ${\tilde{S}}_{i-1,i}$ on the link in between. (c) A filled (empty) bullet denotes an occupied (empty) site, which are mapped by the Jordan-Wigner transformation to the pseudospin up (down) state. (d) The tunneling process of panel (b), expressed in spin language.

Figure 2

Ground states of the QLM in several equivalent representations, for $m/J\to -\infty$ (left) and $m/J\to +\infty$ (right; only one of two degenerate solutions is drawn). Sketches are for vanishing quantum fluctuations ($J=0^{+}$), periodic boundary conditions, and gauge fields represented by spins $1/2$. (a) Staggered-fermion QLM. Physical states are invariant under gauge transformations with generators ${\tilde{G}}_i$, which span one matter and two gauge fields. (b) Associated quark picture. Fermions on even sites correspond to quarks ($q$, light green) and vacancies on odd sites to antiquarks ($\overline{q}$, dark red); the opposite configurations denote the absence of (anti)quarks. For $m/J\to -\infty$, alternating sources and drains of flux completely cover the chain, and the ground state is charge and parity invariant. For $m/J\to +\infty$, the system is empty of matter and has an unbroken flux string, breaking $C$ and $P$ symmetry. In this case, the Gauss law (4) requires the same orientation for all gauge fields, allowing a double energy degeneracy, corresponding to the two directions of electric flux (only one of which is visualized here). (c) Equivalent spin model ${\tilde{H}}_{\mathrm{QLM}}$. Fermions are mapped to spins via the Jordan-Wigner transformation. For $S=1/2$ quantum links, right- (left-) flowing flux translates to an up (down) gauge-field spin. (d) Rotated spin model $H_{\mathrm{QLM}}$ for experimental implementation. At $m/J\to -\infty$, the order parameters are characterized by $\sum _i⟨S_{i,i+1}^z⟩=-1$ and $\sum _i⟨{\sigma }_i^z⟩=1$. In the $C$- and $P$-breaking state at $m/J\to -\infty$, they take the values $\sum _i⟨S_{i,i+1}^z⟩=0$ and $\sum _i⟨{\sigma }_i^z⟩=-1$.

Figure 3

Proposed experimental scheme. (a) Two kinds of pseudospins encode matter and gauge fields in an alternating fashion, with associated spin matrices ${\sigma }_i$ and $S_{i,i+1}$. The corresponding internal states are labeled as $|\downarrow {⟩}_{\sigma }$, $|\uparrow {⟩}_{\sigma }$ and $|\downarrow {⟩}_S$, $|\uparrow {⟩}_S$, respectively. (b) and (c) Spin interactions of $zz$ type are transmitted by coupling the pseudospin levels (sketched for a single ion) to phonons. (b) For hyperfine qubits, one can engineer these couplings via a differential light shift induced by two pairs of Raman beams (with Rabi frequencies ${\Omega }_{1,2}^{\uparrow ,\downarrow }$ and detuning $|\Delta |\gg {\omega }_{\alpha }$; here ${\omega }_{\alpha }$ is the qubit energy difference, with $\alpha =\sigma$, $S$). The Raman transition is detuned by ${\delta }_x$ from the phonon frequency ${\omega }_x$. $|n⟩$ denotes the phonon-number Fock state. (c) For optical qubits, one can use two beams with Rabi frequencies ${\Omega }_1={\Omega }_2={\Omega }^{\prime }$ that are tuned (with a small detuning ${\delta }_x$) close to half the phonon frequency ${\omega }_x$.

Figure 4

Schematic-level diagram of a single ion summarizing the applied spin-phonon couplings (sketched for optical quadrupole qubits as in ${}^{40}\mathrm{Ca}^{+}$). The processes in Figs. 3b, 3c couple ${\sigma }_i^z$ and $S_{i,i+1}^z$ to radial phonon modes. The correct weight of $zz$ interactions for the Gauss law requires effective coupling strengths $|{\Omega }_S|=8|{\Omega }_{\sigma }|$. The $xx$ interactions in $H_K$ may be generated by Mølmer-Sørensen–type couplings of strength ${\Omega }_{\sigma ,xx}$. Coupling to undesired internal levels is avoided by selection rules and Zeeman splittings $\Delta E_{\mathrm{Zee},D/S}$. Whether the ion represents an $S$ or a $\sigma$ spin is determined by its initialization in the state $|\downarrow {⟩}_S$ or $|\downarrow {⟩}_{\sigma }$.

Figure 5

Validity of the microscopic model. (a) and (b) Gauge invariance, quantified by ${\overline{G}}^2\equiv \sum _i{G}_i^2/N_{\mathrm{m}}$, is only violated substantially when $V/J$ is not a strong energy scale. (c) At large $J/V=KB/V^2$, the gauge fields acquire a finite ${\overline{S}}^x\equiv \sum _i{S}_{i,i+1}^x/N_{\mathrm{m}}$. The detrimental effects encountered in panels (a)–(c) are enhanced around $m=0$, i.e., close to the quantum phase transition of the QLM. Curves in panels (a) and (c) are for $m/J\ge 0$, as given in the figures. The behavior for $m/J<0$ is very similar. (d) The overlap $F=⟨\psi (m,J,V)|{\psi }_{\mathrm{ideal}}(m,J)⟩$ between the ground states of the microscopic model and the ideal QLM, Eq. (13), is largest at intermediate $J/V$ and increases with increasing $|m/J|$.

Figure 6

Ground-state behavior of the microscopic model. (a)–(c) Similar to the ideal QLM, the system reaches ${\overline{S}}^z\approx -1$ and ${\overline{\sigma }}^z\approx 1$ for $m/J\to -\infty$, and ${\overline{S}}^z\approx 0$ and ${\overline{\sigma }}^z\approx -1$ for $m/J\to \infty$. This denotes a transition from a parity- and charge-symmetric state to a parity- and charge-symmetry breaking state. Curves from lighter to darker and shorter to longer dashes: $J/V=0.125$, 0.25, 0.275. Solid line: Ideal QLM. The curves change their sign of curvature twice, indicating two phase transitions. (d) Similarly, the fidelity susceptibility $\chi (m,J)$ has two peaks, suggesting two phase transitions. This behavior is in contrast to the ideal model, where $\chi (m,J)$ has a single peak at $m=0$.

Figure 7

Final state after a linear sweep, $m(t)=m_{\mathrm{init}}+t\delta m/\delta t$, with $m_{\mathrm{fin}}=-m_{\mathrm{init}}$ and fixed $V$ and $J$. Solid lines from lighter to darker shades [bottom to top in panel (a)]: $m_{\mathrm{init}}/J=-2$, $-3$, $-4$ for $\delta m/\delta t=0.2V^2/\hslash$. Short (long) dashed lines: $m_{\mathrm{init}}/J=-3$ and $\delta m/\delta t=0.4(0.1)V^2/\hslash$. The arrows show how the precision of the quench increases for larger $m_{\mathrm{init}}$ and smaller quench velocities $\delta m/\delta t$. The total times considered range from $t_{\mathrm{fin}}=15\hslash /V$ (for $m_{\mathrm{init}}/J=-3$, $\delta m/\delta t=0.4V^2/\hslash$) to $60\hslash /V$ ($m_{\mathrm{init}}/J=-3$, $\delta m/\delta t=0.1$). For $V/\hslash =2\pi 1\mathrm{kH}z$ [8, 17], these correspond to 3.2–9.5 ms. (a)–(d) For a value of $J/V\approx 0.15–0.2$, the final, $C$- and $P$-breaking state is reached with (a) high fidelity, (b) small gauge variance, and the desired behavior for (c) the gauge field and (d) the matter field.

Figure 8

Setup for the minimal implementation using four ions. Spin-spin interactions are created by pairs of Raman beams, as explained in Fig. 3, with strengths ${\Omega }_S=2{\Omega }_{\sigma }$. By tuning close to the vibrational mode $q_0$ characterized by amplitudes ${\mathcal{M}}_{n{q}_0}=(1,-1,-1,1{)}_n/2$, one obtains $zz$ spin-spin couplings, as desired for ${\tilde{H}}_G$ [Eq. (10)] (green solid lines). An additional undesired, weaker interaction (orange dashed line) does not affect the system dynamics. Mediated by a different vibrational mode, an additional laser pair generates the $xx$ interaction, as required for $H_K$ (blue dotted line).

Figure 9

Product states at $m/J=\pm \infty$ for the minimal implementation of four ions. (a) State at $m/J\to -\infty$, conserving $C$ and $P$ invariance. (b) State at $m/J\to +\infty$, breaking $C$ and $P$ invariance. This state is degenerate with the opposite configuration of gauge-field spins (i.e., $|S_{12}⟩=|\uparrow ⟩$, $|S_{21}⟩=|\downarrow ⟩$).

Figure 10

Observables in the minimal implementation. (a) The gauge invariance is only violated substantially where $V/J$ is not a strong energy scale. The order parameters (b) ${\overline{S}}^z\equiv (S_{12}^z+S_{21}^z)/2$ and (c) ${\overline{\sigma }}^z\equiv ({\sigma }_1^z+{\sigma }_2^z)/2$ behave similarly to the ideal QLM (solid line). The transition from a $C$- and $P$-invariant state to a $C$- and $P$-breaking state can be clearly seen. The blue curves from lighter to darker and shorter to longer dashes denote $J/V=0.125$, 0.25, 0.275, but practically coincide.

Figure 11

Final state after a linear sweep for the minimal implementation. We change $m(t)=m_{\mathrm{init}}+t\delta m/\delta t$, with $m_{\mathrm{fin}}=-m_{\mathrm{init}}$, and keep $V$ and $J$ fixed. Solid lines from lighter to darker shades: $m_{\mathrm{init}}/J=-2$, $-3$, $-4$ for $\delta m/\delta t=0.2V^2/\hslash$. Short (long) dashed lines: $m_{\mathrm{init}}/J=-3$ and $\delta m/\delta t=0.4(0.1)V^2/\hslash$. The arrows show how the precision of the quench increases for smaller quench velocities $\delta m/\delta t$. The total times considered range from $t_{\mathrm{fin}}=15\hslash /V$ (for $m_{\mathrm{init}}/J=-3$, $\delta m/\delta t=0.4V^2/\hslash$) to $60\hslash /V$ ($m_{\mathrm{init}}/J=-3$, $\delta m/\delta t=0.1$). For $V/\hslash =2\pi$ 1 kHz[8, 17], these correspond to 3.2–9.5 ms. (a)–(d) For a value of $J/V\gtrsim 0.1$, the final $C$- and $P$-breaking state is reached (a) with high fidelity, (b) good gauge invariance, and the desired behavior for (c) the gauge field and (d) the matter field.