Towards analog quantum simulations of lattice gauge theories with trapped ions

Gauge field theories play a central role in modern physics and are at the heart of the Standard Model of elementary particles and interactions. Despite significant progress in applying classical computational techniques to simulate gauge theories, it has remained a challenging task to compute the real-time dynamics of systems described by gauge theories. An exciting possibility that has been explored in recent years is the use of highly controlled quantum systems to simulate, in an analog fashion, properties of a target system whose dynamics are difficult to compute. Engineered atom-laser interactions in a linear crystal of trapped ions offer a wide range of possibilities for quantum simulations of complex physical systems. Here we devise practical proposals for analog simulation of simple lattice gauge theories whose dynamics can be mapped onto spin-spin interactions in any dimension. These include $1+1\mathrm{D}$ quantum electrodynamics, $2+1\mathrm{D}$ Abelian Chern-Simons theory coupled to fermions, and $2+1\mathrm{D}$ pure $Z_2$ gauge theory. The scheme proposed, along with the optimization protocol applied, will have applications beyond the examples presented in this work, and will enable scalable analog quantum simulation of Heisenberg spin models in any number of dimensions and with arbitrary interaction strengths.

Figure 1

A schematic representation of a Raman-beams configuration that induces effective spin-spin interactions in the Heisenberg model. The $N$ sets of individual beams can be chosen along the $(\xi \widehat{X},\xi \widehat{Y},\chi \widehat{Z})$ unit vector ($2{\xi }^2+{\chi }^2=1$). Global beams $\left(I\right)$, $\left(II\right)$, and $\left(III\right)$ are then chosen to propagate along $(-\xi \widehat{X},\xi \widehat{Y},\chi \widehat{Z})$, $(\xi \widehat{X},\xi \widehat{Y},-\chi \widehat{Z})$, and $(\xi \widehat{X},-\xi \widehat{Y},\chi \widehat{Z})$, respectively. These will cause net $\mathrm{\Delta }\mathit{k}$ vectors compared with the individual beams along the $\widehat{X}$, $\widehat{Z}$, and $\widehat{Y}$ directions, respectively. Chosen values of these parameters for the examples of this work are given in Appendix pp1.

Figure 2

(a) The effective spin-spin coupling matrix $J^{\left(xx\right)}$ in Eq. (13) resulting from pairs of Raman beams addressing four individual ions at the Rabi frequency ${\mathrm{\Omega }}^{\left(i\right)}$, where $i=1,...,4$. All beams are detuned from the transverse center-of-mass (c.m.) mode, ${\omega }_1^T=2\pi \times 4.135\text{MHz}$, by the same frequency, ${\mu }_I-{\omega }_1^T=-2\pi \times 830$ kHz. The Lamb-Dicke parameter, $\eta$, multiplying the Rabi frequencies in the figure is $\eta =\sqrt{{\left(\mathrm{\Delta }{k}_I\right)}^2/4\pi M{\nu }^T}\approx 0.068$. (b) With the same detuning, the Rabi frequencies can be adjusted to match the magnitude of the $J_{i,j}^{\left(xx\right)}$ matrix elements for $|j-i|=1$ in (a), producing exactly equal magnitude on these elements, in addition to small non-nearest-neighbor contributions, as shown in (b). Here the $J^{\left(xx\right)}$ matrix is tuned to produce $H^{\left(xx\right)}$ of the four fermion-site Schwinger model in Eq. (19) with $x=6$. Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 3

(a) The effective spin-spin coupling matrix $J^{\left(yy\right)}$ in Eq. (14) resulting from pairs of Raman beams addressing four individual ions at the Rabi frequency ${\mathrm{\Omega }}^{\left(i\right)}$, where $i=1,...,4$. All beams are detuned from the axial c.m. mode, ${\omega }_4^A=2\pi \times 0.713\text{MHz}$, by the same frequency, ${\mu }_{II}-{\omega }_4^A=2\pi \times 3160$ kHz. The Lamb-Dicke parameter, $\eta$, multiplying the Rabi frequencies in the figure is $\eta =\sqrt{{\left(\mathrm{\Delta }{k}_{II}\right)}^2/4\pi M{\nu }^A}\approx 0.081$. (b) With the same detuning, the Rabi frequencies can be adjusted to match the magnitude of the $J_{i,j}^{\left(yy\right)}$ matrix elements for $|j-i|=1$ in (a), producing exactly equal magnitude on these elements, in addition to small $\lesssim 3\%$ non-nearest-neighbor contributions, as shown in (b). Here the $J^{\left(yy\right)}$ matrix is tuned to produce $H^{\left(yy\right)}$ of the four fermion-site Schwinger model in Eq. (20) with $x=6$. Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 4

(a) The effective spin-spin coupling matrix $J^{\left(zz\right)}$ in Eq. (15) resulting from pairs of Raman beams addressing four individual ions at the Rabi frequency ${\mathrm{\Omega }}^{\left(i\right)}$, where $i=1,...,4$. All beams are detuned from the transverse c.m. mode, ${\omega }_1^T=2\pi \times 4.135\text{MHz}$, by the same frequency, ${\mu }_{III}-{\omega }_1^T=2\pi \times 100$ kHz. The Lamb-Dicke parameter, $\eta$, multiplying the Rabi frequencies in the figure is $\eta =\sqrt{{\left(\mathrm{\Delta }{k}_{III}\right)}^2/4\pi M{\nu }^T}\approx 0.068$. (b) With the same detuning, the Rabi frequencies can be adjusted so that the $J^{\left(zz\right)}$ matrix produces the long-range couplings in $H^{\left(zz\right)}$ of the four fermion-site Schwinger model in Eq. (21) with $x=6$. Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 5

The effective magnetic field on each ion, $B_z^{\left(i\right)}$, that produces the $H^{\left(z\right)}$ Hamiltonian of the Schwinger model, Eq. (22), for $N=4$ and $\mu =1$. Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 6

(a) Time evolution of the state $|{\psi }^{\left(\text{vac}\right)}\rangle =|\downarrow \uparrow \downarrow \uparrow \rangle$ corresponding to the strongly interacting vacuum of the four fermion-site lattice Schwinger model with $x=0.6$ and $\mu =0.1$. Open circles in the plots in (b)–(e) are the values of VPA at select times with 20 inexact $H^{\left(xx\right)}(=H^{\left(yy\right)})$ Hamiltonians, as listed in the Supplemental Material [83]. The nine data points that satisfy ${\mathrm{\Delta }}^{\left(xx\right)}\equiv {J_{1,3}^{\left(xx\right)}}^2+{J_{1,4}^{\left(xx\right)}}^2+{J_{2,4}^{\left(xx\right)}}^2\le {10}^{-4}$ are chosen to define central values (dark-pink lines) and uncertainties (pink bands) on the VPA, and are compared with the exact expectations (blue lines). The plot in (a) represents the exact time evolution of vacuum (blue curve) compared with the central value (dark-pink curve) and uncertainty (pink band) on the VPA obtained from nine Hamiltonians that give rise to ${\mathrm{\Delta }}^{\left(xx\right)}\le {10}^{-4}$. Numerical values associated with these plots are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 7

(a) The effective spin-spin coupling matrix $J^{\left(zz\right)}$ in Eq. (15) resulting from multiple pairs of Raman beams addressing $N=8$ individual ions at the Rabi frequencies ${\mathrm{\Omega }}_{III,m^{\prime }}^{\left(i\right)}$ shown in (b), where $i=1,...,8$ and $m^{\prime }=1,...,7$. The pairs of beams addressed at ion $i$ are detuned from the transverse c.m. mode by seven different frequencies, ${\mu }_{III,m^{\prime }}={\omega }_{m^{\prime }}^T+f_s({\omega }_{m^{\prime }}^T-{\omega }_{m^{\prime }+1}^T)$ with $f_s=-0.5$, as denoted in (c). The Lamb-Dicke parameter, $\eta$, multiplying the Rabi frequencies in the figure is $\eta =\sqrt{{\left(\mathrm{\Delta }k\right)}^2/4\pi M{\nu }^T}\approx 0.068$. Here the $J^{\left(zz\right)}$ matrix is tuned to produce $H^{\left(zz\right)}$ of the eight fermion-site Schwinger model in Eq. (21). Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 8

(a) A $4\times 4$ lattice of spins ($s=\frac{1}{2}$) with nearest-neighbor interactions, corresponding to the ${\sigma }_x\otimes {\sigma }_x$ (or equivalently ${\sigma }_y\otimes {\sigma }_y$) interactions in the Hamiltonian in Eq. (26) with $\mathit{n}=(n_x,n_y)$, where $n_x$ ($n_y$) runs from 0 to 3, and where an open boundary condition is adopted. The nearest-neighbor interactions of a select site are depicted in green links. (b) This 2D configuration can be mapped to a 1D chain of ions, along with the couplings of the select site in the new configuration. The obtained 1D coupling matrix $J_{i,j}$ is shown in (c).

Figure 9

(a) A $5\times 5$ spatial lattice corresponding to the $Z_2$ Hamiltonian in Eq. (27). An open boundary condition is adopted, and a select plaquette term in the Hamiltonian is shown. The center of the plaquettes defines the sites of a dual lattice, as depicted by the green points, and are separately shown in (b). Such a 2D configuration corresponds to the Ising Hamiltonian in Eq. (28), which can now be mapped to a 1D chain of ions, as shown in (c).

Figure 10

The level diagram of ${}^{171}\mathrm{Yb}^{+}$ relevant to the scheme presented in this Appendix.

Figure 11

(a) The effective spin-spin coupling matrix $J^{\left(xx\right)}$ in Eq. (13) resulting from multiple pairs of Raman beams addressing $N=8$ individual ions at the Rabi frequencies ${\mathrm{\Omega }}_{I,m^{\prime }}^{\left(i\right)}$ shown in (b), where $i=1,...,8$ and $m^{\prime }=1,...,7$. The pairs of beams addressed at ion $i$ are detuned from the transverse c.m. mode by seven different frequencies, ${\mu }_{I,m^{\prime }}={\omega }_{m^{\prime }}^T+f_s({\omega }_{m^{\prime }}^T-{\omega }_{m^{\prime }+1}^T)$ with $f_s=0.5$, as denoted in (c). The Lamb-Dicke parameter, $\eta$, multiplying the Rabi frequencies in the figure is $\eta =\sqrt{{\left(\mathrm{\Delta }{k}_I\right)}^2/4\pi M{\nu }^T}\approx 0.068$. Here the $J^{\left(xx\right)}$ matrix is tuned to produce $H^{\left(xx\right)}$ of the eight fermion-site Schwinger model in Eq. (19) with $x=6$. Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 12

(a) The effective spin-spin coupling matrix $J^{\left(yy\right)}$ in Eq. (14) resulting from multiple pairs of Raman beams addressing $N=8$ individual ions at the Rabi frequencies ${\mathrm{\Omega }}_{II,m^{\prime }}^{\left(i\right)}$ shown in (b), where $i=1,...,8$ and $m^{\prime }=1,...,7$. The pairs of beams addressed at ion $i$ are detuned from the axial c.m. mode by seven different frequencies, ${\mu }_{I,N-m^{\prime }+1}={\omega }_{N-m^{\prime }+1}^A+f_s({\omega }_{N-m^{\prime }}^A-{\omega }_{N-m^{\prime }+1}^A)$ with $f_s=-0.5$, as denoted in (c). The Lamb-Dicke parameter, $\eta$, multiplying the Rabi frequencies in the figure is $\eta =\sqrt{{\left(\mathrm{\Delta }{k}_{II}\right)}^2/4\pi M{\nu }^A}\approx 0.081$. Here the $J^{\left(yy\right)}$ matrix is tuned to produce $H^{\left(yy\right)}$ of the eight fermion-site Schwinger model in Eq. (20) with $x=6$. Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 13

The effective magnetic field, $B_z$, that produces the $H^{\left(z\right)}$ Hamiltonian of the Schwinger model, Eq. (22), for $N=8$ and $\mu =1$. Numerical values associated with this figure are provided in the Supplemental Material [83]. These values are obtained from a numerical study and do not represent experimental findings.

Figure 14

Contributions to the exponent of the full laser-ion evolution operator up to and including $O({\eta }^2,\eta B)$ for laser parameters found in the single-frequency, multiamplitude scheme in Sec. 3 to engineer the four fermion-site Schwinger Hamiltonian with $x=6$ and $\mu =1$. The quantities plotted are enumerated in this Appendix and are dimensionless. The horizontal axis is time in ms. The plots shown correspond to the evolution of the first ion in the chain. The results for the rest of the ions can be found in the Supplemental Material [83].

Figure 15

Contributions to the exponent of the full laser-ion evolution operator up to and including $O({\eta }^2,\eta B)$ for laser parameters found in the multifrequency, multiamplitude scheme in Sec. 3 to engineer the eight fermion-site Schwinger Hamiltonian with $x=6$ and $\mu =1$. The quantities plotted are enumerated in this Appendix and are dimensionless. The horizontal axis is time in ms. The plots shown correspond to the evolution of the first ion in the chain. The results for the rest of the ions can be found in the Supplemental Material [83].