Analog quantum simulation of $(1+1)$-dimensional lattice QED with trapped ions

The prospect of quantum-simulating lattice gauge theories opens exciting possibilities for understanding fundamental forms of matter. Here, we show that trapped ions represent a promising platform in this context when simultaneously exploiting internal pseudospins and external phonon vibrations. We illustrate our ideas with two complementary proposals for simulating lattice-regularized quantum electrodynamics (QED) in $(1+1)$ space-time dimensions. The first scheme replaces the gauge fields by local vibrations with a high occupation number. By numerical finite-size scaling, we demonstrate that this model recovers Wilson's lattice gauge theory in a controlled way. Its implementation can be scaled up to tens of ions in an array of microtraps. The second scheme represents the gauge fields by spins $\frac{1}{2}$, and thus simulates a quantum link model. As we show, this allows the fermionic matter to be replaced by bosonic degrees of freedom, permitting small-scale implementations in a linear Paul trap. Both schemes work on energy scales significantly larger than typical decoherence rates in experiments, thus enabling the investigation of phenomena such as string breaking, Coleman's quantum phase transition, and false-vacuum decay. The underlying ideas of the proposed analog simulation schemes may also be adapted to other platforms, such as superconducting qubits.

Figure 1

$(1+1)\mathrm{D}$ QED on a lattice and truncation schemes. (a) The lattice representation of $(1+1)\mathrm{D}$ QED considers a system of free fermions living on sites $\left({\psi }_i\right)$, coupled to a U(1) Abelian gauge field living on the links connecting neighboring sites $\left(E_{i.i+1},U_{i,i+1}\right)$. In the standard Wilson LGT, the gauge-field degrees of freedom on each link span a 2D quantum rotor Hilbert space. (b) We introduce here a complementary LGT, the highly occupied boson model, by representing the gauge fields by local boson modes $\left(a_{i,i+1},a_{i,i+1}^{†}\right)$, illustrated as harmonic oscillators. In the limit of large occupation number, $N\to \infty$, the Wilson LGT is recovered exactly. (c) An alternative scheme, the quantum link model [7, 8], represents the gauge fields by spin operators $\left(s_{i,i+1}^z,s_{i,i+1}^{+}\right)$. It approaches the Wilson formulation in the limit of large spins, $S\to \infty$. (d) In one spatial dimension, the lattice fermions can be mapped to spins via the Jordan-Wigner transformation. They represent the Dirac spinor of charges $\left(e^{-}\right)$ and anticharges $\left(e^{+}\right)$ in a staggered fashion; i.e., on an even (odd) site, the presence (absence, denoted by Ø in the figure) of a staggered fermion corresponds to the presence of $e^{-}$. (e) The gauge-field Hilbert space on a single link, from top to bottom for $(1+1)\mathrm{D}$ QED, the highly occupied boson model, and the quantum link model.

Figure 2

String-breaking dynamics on a lattice with $L=12$ sites, as a quantitative measure of how reliably the HOBM approximates the lattice QED. Parameters are $\mu =V=0.2J$, and the initial state is a charge-anticharge pair at the boundary of the lattice connected by a string of electric field. (a) The electric-field string breaks by spontaneous creation of charge-anticharge pairs, as indicated by a reduction of the absolute value of the space-averaged electric-field strength, $\langle E\left(t\right)\rangle$. The HOBM (light solid line and squares) manifests similar string-breaking dynamics to that of lattice QED and reaches quantitative agreement over the considered times for moderate $N$. After additional time averaging (dashed), the curves for both models are hard to discern. (b) The time-averaged error, $\epsilon \left(t\right)$, quantifying the difference between the HOBM and lattice QED, remains bounded during time evolution and is suppressed by higher boson number $N$. (c) Also under varying the matter–gauge-field coupling $J,\epsilon \left(t\right)$ remains always bounded (here shown at $t=40\pi /J$ as an example), and is suppressed with larger boson number $N$.

Figure 3

Quantitative analysis of Coleman's parity-symmetry-breaking phase transition at vacuum angle $\pi$, for both lattice QED and the HOBM. The coupling parameter is chosen as $ga=0.3$, which locates the quantum critical point of lattice QED at ${(m/g)}_{\mathrm{c}}=0.297$ [47]. (a) The phase transition is quantified by the order parameter $\langle E\rangle$, with a slight shift between the HOBM and lattice QED. (b), (c) Finite-size scaling for (b) the order parameter at criticality and (c) the position of the pseudocritical point. Boson occupation number $N\in [10,200]$, while curves from light to dark blue denote increasing lattice sites $L=10,12,14,16,18$. The rescaled data collapse onto a single curve, suggesting that the HOBM has the same quantum critical behavior as lattice QED, as long as $N\to \infty$ faster than $L\to \infty$.

Figure 4

Engineering of localized radial phonon modes in a string of microtraps. (a) The radial trapping frequencies ${\omega }_{x,y}$ increase by ${\mathrm{\Delta }}_T$ along the chain in a stepwise fashion, with an offset of one lattice site between $x$ (blue) and $y$ (green). The large trapping-frequency mismatch ${\mathrm{\Delta }}_T$ effectively localizes hybridized phonon modes within pairs of ions. Phonon modes $c_l^{\alpha }$ and $c_{l+1}^{\alpha }$ are shared by ions $l$ and $l+1$ (with $\alpha =x$ for $l=\mathrm{odd}$ and $\alpha =y$ for $l=\mathrm{even})$. In each pair, the phonon mode $c_l^{\alpha }$ is chosen to encode the bosonic gauge field, while the residual phonon $c_{l+1}^{\alpha }$ serves as quantum data bus to transmit interactions. (b) Calculation of the matrix $M_{lq}^{x\left(y\right)}$, which diagonalizes the vibrations, for realistic parameters $({}^9{\mathrm{Be}}^{+}$ ions, with the trap parameters ${\mathrm{\Delta }}_T=2\pi \times 500$ kHz, ${\delta }_T=2\pi \times 5$ kHz, ${\omega }_x={\omega }_y=2\pi \times 5$ MHz, and distance between trap centers $d=30\mu \mathrm{m})$. The leakage of $M_{lq}^{\alpha }$ out of pairs of ions is well below 0.01 along both $x$ and $y$ directions, showing that the phonons are efficiently localized in pairs of ions. (c) Since the trapping frequency cannot be increased infinitely, scaling to long ion chains requires repetition of elementary blocks containing a few pairs of ions [such as depicted in panel (a)]. A global trapping-frequency offset ${\mathrm{\Delta }}_B$ between each block eliminates long-range interblock hopping of the localized phonons. (d) For realistic parameters, the leak-out of vibrational modes outside of the desired pair remains well below 0.01 even for a large number of blocks. Here, ${\mathrm{\Delta }}_B=2\pi \times 50$ kHz, with other parameters the same as in panel (b); block size is 6 ions, and the number of blocks is in the range of 1–10. The leak-out is measured by the maximum element of $|M^{\alpha }-M_0^{\alpha }|$, where $M^{\alpha }$ is the normal-mode distribution matrix in the $\alpha$ direction, and the block diagonal $M_0^{\alpha }$ is its zeroth-order approximation. Two situations are considered: the ideal case, where the radial trapping frequencies are designated by Eqs. (18) and (19), and the nonideal case, where a local randomness of the trapping frequencies, uniformly distributed in $2\pi \times [-30,30]$ kHz, is added on top of Eqs. (18) and (19), to reflect the uncertainties in the frequency control in experiments.

Figure 5

Design of the HOBM Hamiltonian Eq. (6). (a) The engineering is independent in each basic element consisting of two neighboring ions $l$ and $l+1$. The design of the spin–gauge-field interaction $\propto J$ involves the pseudospins with states $\{{|g\rangle }_l,{|e\rangle }_l\}$ and $\{{|g\rangle }_{l+1},{|e\rangle }_{l+1}\}$, as well as the local phonon modes $c_l^{\alpha }$ and $c_{l+1}^{\alpha }$, and uses two local laser beams ${\mathrm{\Omega }}_l^{\alpha }$ and ${\mathrm{\Omega }}_{l+1}^{\alpha }$ (with $\alpha =x$ for $l=2n-1=\mathrm{odd}$ and $\alpha =y$ for $l=2n=\mathrm{even})$. The creation of the gauge-field energy term $\propto V$ relies on a strongly detuned standing-wave radiation (red sinusoid) to the first ion in the element, applied in the $x\left(y\right)$ direction depending on $l=\mathrm{odd}\left(\mathrm{even}\right)$. (b) Generation of the spin–gauge-field coupling illustrated for the element consisting of ions 1 and 2. ${\mathrm{\Omega }}_1^x$ drives the first ion on the second blue sideband consisting of the excitation of one gauge-field phonon $\left(c_1^x\right)$ and one data-bus phonon $\left(c_2^x\right)$, while ${\mathrm{\Omega }}_2^x$ drives the second ion on the first blue sideband of the data-bus phonon $c_2^x$. The detuning $\delta$ is chosen far larger than the sideband-transition strengths, so the resonant process is a flip-flop transition between the two pseudospins. (c) Creation of the gauge-field energy term $\propto V$, exemplified for the first element, where a standing wave acts on ion 1 along the $x$ direction. Here, we consider the example of ${}^9{\mathrm{Be}}^{+}$ ions, where the pseudospin is encoded by the two hyperfine levels $(F,m_F)=(1,1)$ and $(F,m_F)=(2,2)$ of the ${}^2{S}_{1/2}$ manifold. The standing wave drives off-resonantly the optical transitions from both spin states to the ${}^2{P}_{1/2}$ manifold, with detuning ${\mathrm{\Delta }}_1^{\mathrm{sw}}$ and Rabi frequency ${\mathrm{\Omega }}_1^{\mathrm{sw}}$. For large enough ${\mathrm{\Delta }}_1^{\mathrm{sw}}$, it creates nearly equal ac Stark shifts to both spin-up and spin-down states, as discussed in detail in Sec. 3a3.

Figure 6

Numerical predictions for string breaking and Coleman's quantum phase transition on a modest-size quantum simulator, built according to the HOBM scheme proposed in Sec. 3, compared to that of the ideal HOBM and $(1+1)\mathrm{D}$ QED. (a) String-breaking dynamics in a system of $L=12$ ions, over a relatively long evolution period $t\in [0,40\pi /J]$, to benchmark the imperfections induced by the ac Stark shifts. The experimental parameters are chosen so that $J=2\pi \times 120$ Hz, $V=\mu =2\pi \times 25$ Hz, and the local radial phonon modes which encode the gauge-field DOFs are prepared into an initial Fock state with occupation number $N=10$. (b), (c) Similar to the case of the ideal HOBM, Fig. 2, the time-averaged error for the quantum simulator, $\epsilon \left(t\right)={\int }_0^{t}d{t}^{\prime }|{\langle E\left(t^{\prime }\right)\rangle }_{\mathrm{QS}}-{\langle E\left(t^{\prime }\right)\rangle }_{\mathrm{QED}}|/t$, is bounded during time evolution and is suppressed with increasing boson number $N$. The error is dominated by the ac Stark shifts, which however only induce small deviations from the ideal results and do not break the Gauss law. (d) Comparison of the critical behavior of the order parameter $\langle E\rangle$, between the quantum simulator and the ideal HOBM, in the Coleman's quantum phase transition at vacuum angle $\pi$. We choose $J=2\pi \times 120$ Hz and $V=2\pi \times 25$ Hz, yielding $g=2\sqrt{JV}\simeq 2\pi \times 55$ Hz. By changing the small detuning $\mu =m$, we scan across the quantum critical point, and find that the quantum simulator represents the critical behavior of the HOBM faithfully.

Figure 7

The energy lattice scheme for the quantum simulation of a $S=\frac{1}{2}$ U(1) quantum link model in $(1+1)$ space-time dimensions. (a) In this scheme, the $S=\frac{1}{2}$ gauge link variable $s_{i,i+1}$ is encoded in the internal pseudospin states of the $i\mathrm{th}$ ion, ${|g\rangle }_i$ and ${|e\rangle }_i$. The matter field $c_i$ is realized by the collective phonon mode $c_q^z$ with same index number, ordered by energy ${\epsilon }_q^z$. The Gauss law, implemented by initial-state preparation, forbids double excitation of the phonons, thus rendering the bosonic model a valid $S=\frac{1}{2}$ QLM. (b) The gauge-field–matter interaction can be realized by near-resonant second red-sideband transitions on individual ions, which can be implemented straightforwardly with single-ion addressability.

Figure 8

Numerical comparison of the dynamics after a quench, between the ideal $S=\frac{1}{2}$ QLM and a small-scale quantum simulation according to the energy lattice scheme proposed in Sec. 4. The lattice size is chosen as $L=6$, and the matter–gauge-field coupling strength $J=2\pi \times 500$ Hz. The system is initially prepared in a parity-symmetry-broken ground state at large $\mu$. For $t>0$ it evolves under the Hamiltonian at $\mu =0$, which favors a parity-symmetry-restored ground state. (a) Coherent oscillations of the order parameter $\langle E\rangle ={\sum }_l{(-1)}^l{\sigma }_l^z/L$. The carrier-transition light shift, $H_{\mathrm{ls}0}$, if not compensated, induces significant errors (orange dotted curve). If $H_{\mathrm{ls}0}$ is compensated by a local adjustment of the detuning of the driving frequencies, the quantum simulator (blue dashed curve) simulates the quantum link model (red solid curve) well, with small gauge-invariant errors induced by light shifts from first-sideband transitions. (b) Evolution of the order parameter for the quantum simulator, taken from (a) but with amplified resolution. The fast, small-amplitude oscillations are due to off-resonant transitions to other sidebands. They have negligible influence on the quantum simulation. (c) The Gauss law remains intact to an extremely high degree during the entire quench dynamics, since the main errors due to ac Stark shifts are gauge invariant. The insignificant violation of the Gauss law is due to small off-resonant transitions to other sidebands, as seen in panel (b).

Figure 9

Compensation scheme of the ac Stark shifts, here shown for the element $(1,2)$ as an example. An additional laser beam, with parameters $({\mathrm{\Omega }}_l^{\mathrm{c}},{\omega }_l^{\mathrm{Lc}})$, is applied to the first ion. It is off-resonant to any transition inside the ion pair $(1,2)$; nevertheless it induces an appropriate amount of ac Stark shift, which compensates the part of the light shift quadratic on the gauge-field phonon number induced by ${\mathrm{\Omega }}_1^x$. Similarly, a laser beam with parameter $({\mathrm{\Omega }}_2^{\mathrm{c}},{\omega }_2^{\mathrm{Lc}})$ is applied to ion 2, to compensate part of the light shift linearly dependent on the gauge-field phonon number. The residual ac Stark shifts can be compensated largely by local adjustment of the frequencies of the two sideband laser beams ${\omega }_1^{\mathrm{L}x}$ and ${\omega }_2^{\mathrm{L}x}$.