Floquet-engineered vibrational dynamics in a two-dimensional array of trapped ions

TL;DR

This paper demonstrates Floquet engineering in a scalable 2D trapped-ion system, enabling control over phonon flow and paving the way for simulating topological and gauge phenomena in quantum systems.

Contribution

It introduces a method to control ion interactions and phonon dynamics via local parametric modulations in a 2D ion array, advancing quantum simulation capabilities.

Findings

Controlled phonon trajectories and interferences achieved.

Demonstrated tunable long-range ion couplings.

Opened pathways for simulating topological phenomena.

Abstract

We demonstrate Floquet engineering in a basic yet scalable 2D architecture of individually trapped and controlled ions. Local parametric modulations of detuned trapping potentials steer the strength of long-range inter-ion couplings and the related Peierls phase of the motional state. In our proof-of-principle, we initialize large coherent states and tune modulation parameters to control trajectories, directions and interferences of the phonon flow. Our findings open a new pathway for future Floquet-based trapped-ion quantum simulators targeting correlated topological phenomena and dynamical gauge fields.

Full text

In this Letter, we show essential features of phonon assisted coupling of individual atomic ions trapped at micro sites of our triangular two-dimensional trap array. Parametric drives applied to single or multiple locations steer constructive and destructive coherent couplings within the array. Tuning driving amplitudes and relative phases, we control directionality and interference of the phonon flow via the related synthetic Peierls phase of the motional states, a key requirement for future quantum Floquet engineering.

We trap magnesium ions in our surface-electrode trap array featuring separate micro sites $\text{T}_{\bf j}$, where ${\bf j}\in\left\{0,1,2\right\}$ labels the corners of the triangle with side lengths of $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and an ion-surface distance of $\simeq$40\text{,}\mathrm{\SIUnitSymbolMicro m} Mielenz *et al.* ([2016](#bib.bib23)); [sup ](#bib.bib24). In harmonic approximation, these distances yield an inter-site coupling strength $\Omega_{\text{C}}/(2\pi)\simeq$1\text{\,}\mathrm{kHz} for motional frequencies $\omega_{{\bf j}}/(2\pi)\simeq$4\text{,}\mathrm{MHz}. Heating rates of $110\text{\,}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{/}\mathrm{m}\mathrm{s}$ are derived from calibration measurements close to the motional ground state Kalis ([2017](#bib.bib25)); Friedenauer *et al.* ([2006](#bib.bib26)). In order to receive unambiguous signals of the phonon dynamics we initialize coherent states (exceeding $\simeq$1,000\text{\,}\ \mathrm{m}\mathrm{o}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\ \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}). The effective coupling rate $\Omega_{\text{C,eff}}$ is tunable via relative motional mode orientations and the detuning of the individual trapping sites. Anharmonic contributions of the trapping potential lead to an increased $\Omega_{\text{C,eff}}/(2\pi)\simeq$6\text{,}\mathrm{kHz}, while the related efficiency is reduced accordingly Hakelberg *et al.* ([2018](#bib.bib11)). Quasi-static control potentials locally tune electric fields, curvatures and higher order terms, e.g. for preparation/detection or inter-site couplings Hakelberg *et al.* ([2018](#bib.bib11)). We adiabatically ramp these control potentials within $t_{\text{ramp}}\leq$100\text{\,}\mathrm{\SIUnitSymbolMicro s} ($t_{\text{ramp}}\geq(\omega_{\bf j})^{-1}$) between different configurations. Additionally, we can apply various local periodic control potentials ${\phi^{\text{osc}}_{\bf j}}({\Omega^{\text{osc}}_{\bf j}},{\varphi^{\text{osc}}_{\bf j}},{u^{\text{osc}}_{\bf j}})$ for duration $t^{\text{osc}}_{\bf j}$, oscillating with frequency $\Omega^{\rm osc}_{\bf j}$ with a tunable phase $\varphi^{\text{osc}}_{\bf j}$ and amplitude $u^{\text{osc}}_{\bf j}$, allowing for: (i) Floquet engineering by a parametric modulation {\phi^{\text{\mathcal{M}}}_{\bf j}} of motional frequencies with \Omega^{\text{\mathcal{M}}}_{\bf j}/(2\pi)\simeq100\text{,}\mathrm{kHz} or *(ii)* Initialization of a vibronic coherent state at a given $\text{T}_{\bf j}$ by a local excitation via $\phi^{\mathcal{E}}_{\bf j}$. All experiments are initialized by global Doppler cooling, aligning the local mode orientations, and tuning the lowest motional frequencies to $\omega_{{\bf j}}/(2\pi)\simeq$35\text{\,}\mathrm{MHz} Mielenz et al. (2016); Hakelberg et al. (2018). In following steps (see below) we apply dedicated control potentials for individual settings, and finish by local fluorescence detection allowing to derive average phonon numbers $\bar{\mathrm{n}}_{\bf j}$. The reduction of the fluorescence rate allows us to derive the increase of $\bar{\mathrm{n}}_{\bf j}$. Each sequence is repeated 200 to 400 times to derive the standard error of the mean (s.e.m.).

To calibrate Floquet engineering via oscillating potentials \phi^{\text{\mathcal{M}}}_{\bf j}, see sup , we perform measurements with single ions at $\text{T}_{\bf j}$. Exemplarily, we discuss results for site $\text{T}_{\text{1}}$, see Fig. 1, where we probe the effect of \phi^{\text{\mathcal{M}}}_{1}(\Omega^{\text{\mathcal{M}}}_{\text{1}},u^{\text{\mathcal{M}}}_{\text{1}}) for fixed \Omega^{\text{\mathcal{M}}}_{\text{1}}/(2\pi)=100\text{,}\mathrm{k}\mathrm{H}\mathrm{z}$\ll\omega_{\text{1}}/(2\pi)$ with a simultaneously applied drive for coherent excitation \phi^{\mathcal{E}}_{1}(\Omega^{\text{\mathcal{E}}}_{\text{1}},u^{\text{\mathcal{E}}}_{\text{1}}). Tuning \Omega^{\text{\mathcal{E}}}_{\text{1}}  across ${\omega_{\text{1}}}$, we show reconstructed motional amplitudes for u^{\text{\mathcal{M}}}_{\text{1}}=0\text{,}\mathrm{V} (top), $150\text{\,}\mathrm{mV}$ (middle), and $250\text{\,}\mathrm{mV}$ (bottom) in dependence on $\Omega^{\text{$\mathcal{E}$}}_{\text{1}}$. When $\phi^{\text{$\mathcal{M}$}}_{1}$ is switched on, a comb structure is spanned by several channels (sidebands) at $\Omega^{\mathcal{E}}_{1}\simeq\omega_{1}+m\Omega^{\mathcal{M}}_{1}$ with $m\in\mathbb{Z}$. Floquet theory (solid lines) predicts that the relative strength of these channels is defined by the $m$-th order Bessel function of the first kind $\mathcal{J}_{m}(\eta_{\bf j})$, where $\eta_{\bf j}\propto u^{\text{$\mathcal{M}$}}_{\bf j}/\Omega^{\text{$\mathcal{M}$}}_{\bf j}$ represents the modulation index [sup ](#bib.bib24). For increasing $u^{\text{$\mathcal{M}$}}_{\text{1}}$ (Fig. 1(b) middle, bottom), the carrier channel decreases, until $\mathcal{J}_{0}(\eta_{\bf j})$ crosses zero. This channel gets effectively shut and can lead in following experiments where we couple neighboring sites to so-called coherent destruction of tunneling Grifoni and Haenggi ([1998](#bib.bib27)). Overall, we find $\eta_{\bf j}\Omega^{\text{$\mathcal{M}$}}_{\bf j}/(2\pi)$ of up to $300\text{\,}\mathrm{kHz}$ for $\Omega^{\text{$\mathcal{M}$}}_{\bf j}/(2\pi)\simeq$50200\text{\,}\mathrm{kHz}.

To demonstrate control of synthetic Peierls phases imprinted on the motional state in real-time, we explore the assisted transfer of energy, i.e. flow of phonons, between ions at different $\text{T}_{\bf j}$. As an example we perform our experiment with single ions at $\text{T}_{\text{0}}$ and $\text{T}_{\text{1}}$, see Fig. 2. We adjust the inter-site detuning to $\Delta\omega_{01}/(2\pi)=(\omega_{\text{1}}-\omega_{\text{0}})/(2\pi)\simeq$100\text{,}\mathrm{kHz}, and excited the ion at $\text{T}_{\text{0}}$  ($\phi^{\mathcal{E}}_{0}$ for $t^{\text{$\mathcal{E}$}}_{\text{0}}=$20\text{\,}\mathrm{\SIUnitSymbolMicro s}) to $\bar{\mathrm{n}}_{\text{0}}=$5,00010,000\text{,}$$. By choice of $\Delta\omega_{01}$ the coupling efficiency (for $\eta_{\bf j}=0$) is suppressed by four orders of magnitude. Setting \Omega^{\text{\mathcal{M}}}_{\text{1}}=\Delta\omega_{01}, $\eta_{\text{1}}\simeq 1.8$, at a fixed $\varphi^{\mathcal{M}}_{\text{1}}$, the phonon exchange is enabled by assistance of \phi^{\text{\mathcal{M}}}_{1}: a single transmission channel is opened by overlapping the lower first sideband at $\text{T}_{\text{1}}$ with the carrier at $\omega_{\text{0}}$ of $\text{T}_{\text{0}}$, see Fig. 2(b, inset). In Figure 2(b), we show reconstructed $\bar{\mathrm{n}}_{\text{1}}$ as a function of t^{\text{\mathcal{M}}}_{\text{1}}. We model the coherent exchange, absorption and emission, of phonons, with corresponding assisted coupling rate $\Omega_{\text{AC}}$ (solid line, see sup ). We thus confirm that the ion at $\text{T}_{1}$ absorbs up to $\simeq 820$ phonons after $t_{\pi}=\pi/\Omega_{\text{AC}}$. The efficiency is limited to about $15\text{\,}\mathrm{\char 37\relax}$ of $\bar{\mathrm{n}}_{\text{0}}$. We attribute that to the anharmonicity of the trapping potential probed by the currently large $\bar{\mathrm{n}}_{\text{1}}$ Hakelberg et al. (2018). The anharmonic effects can be interpreted as additional detuning, increasing $\Omega_{\text{AC}}$ but limiting efficiency.

The Peierls phase $\Phi^{\rm P}(t)$ plays an important role in the Floquet engineered Hamiltonian of energy transfer between different sites sup . It is given by the path integral along $l$ between trapping sites $\Phi^{\rm P}(t)=\frac{q}{\hbar}\int_{\text{T}_{\bf j}}^{T_{\bf i}}{\rm d}\boldsymbol{l}\cdot\boldsymbol{A}(\boldsymbol{r},t)$, where $\hbar$ is the reduced Planck constant. $\Phi^{\rm P}$ rules the dynamics of the phonons as if they were particles with charge $q$ coupled to a gauge potential $\boldsymbol{A}(\boldsymbol{r},t)$. In an extended experimental sequence the evolution of $\Phi^{\rm P}$ is controlled in real-time during the experiment. In particular, after a period of assisted coupling of t^{\text{\mathcal{M}}}_{\text{1,prep}}=t_{\pi}/2\simeq150\text{,}\mathrm{\SIUnitSymbolMicro s}, we adiabatically ramp $\varphi^{\mathcal{M}}_{\text{1}}\to\varphi^{\mathcal{M}}_{\text{1}}+\Delta\varphi^{\text{$\mathcal{M}$}}_{\text{1}}$ within $t_{\text{$\varphi$,ramp}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro s}$\gg\omega_{\bf j}^{-1}$, and continue the modulation for duration $t^{\text{$\mathcal{M}$}}_{\text{1}}$-($t^{\text{$\mathcal{M}$}}_{\text{1,prep}}$+$t_{\text{$\varphi$,ramp}}$). We depict normalized $\bar{\mathrm{n}}_{\text{1}}$ as a function of $\Delta\varphi^{\text{$\mathcal{M}$}}_{\text{1}}$ and $t^{\text{$\mathcal{M}$}}_{\text{1}}$ in Fig. 2(c). The dependence of the number of exchanged phonons on $\Delta\varphi^{\text{$\mathcal{M}$}}_{\text{1}}$  shows that $\Phi^{\rm P}$ cannot be simply gauged away for $\Delta\varphi^{\text{$\mathcal{M}$}}_{\text{1}}\neq 0$ but, instead, it serves to control the directionality of the coherent energy flow. At $t^{\text{$\mathcal{M}$}}_{\text{1}}\simeq$300\text{\,}\mathrm{\SIUnitSymbolMicro s}, optimal exchange is observed for \Delta\varphi^{\text{\mathcal{M}}}_{\text{1}}\simeq+\pi/4. For \Delta\varphi^{\text{\mathcal{M}}}_{\text{1}}\approx-3\pi/4, a change by $\pi$, we observe the transferred population nearly vanishing. This is consistent with Floquet theory, which predicts that $\Omega_{\text{AC}}(\varphi^{\mathcal{M}}_{\text{1}})\to-\Omega_{\text{AC}}(\varphi^{\mathcal{M}}_{\text{1}}+\pi)$, equivalent to a time-reversal operation. That is, it returns phonons from $\text{T}_{\text{1}}$  back to $\text{T}_{\text{0}}$ and further simulates the application of an electric field on charged particles. In particular, $\partial\Phi^{\rm P}(t)/\partial t\neq 0$ can be understood as a background synthetic electric field $\boldsymbol{E}(\boldsymbol{r},t)=-\partial\boldsymbol{A}(\boldsymbol{r},t)/\partial t$. We observe a global shift by $\pi/4$ in the data. Numerical simulations can provide evidence for a similar shift, when considering a mismatch between \Omega^{\text{\mathcal{M}}}_{\text{1}} and $\Delta\omega_{01}$ of a few percent and the finite ramping duration t_{\text{\varphi,ramp}}.

In the next sequence, we explore the dynamics between $\text{T}_{\text{0}}$ and $\text{T}_{\text{1}}$ when both sites are locally driven by \phi^{\text{\mathcal{M}}}_{0} and \phi^{\text{\mathcal{M}}}_{1} at fixed \Omega^{\text{\mathcal{M}}}_{\text{0}}=\Omega^{\text{\mathcal{M}}}_{\text{1}}\simeq\Delta\omega_{01} and $\eta_{\text{0}}\simeq\eta_{\text{1}}$. In this case, the assisted exchange can be controlled by the relative modulation phase \Delta\varphi^{\text{\mathcal{M}}}_{\text{01}}=\varphi_{0}^{\mathcal{M}}-\varphi^{\mathcal{M}}_{\text{1}}, here reaching $\Omega_{\text{AC}}/(2\pi)\leq$4.5\text{,}\mathrm{kHz}$$ Hakelberg et al. (2018). As shown in Fig. 3,(b,c inset), we open several transmission channels, and the overall phonon exchange is governed by constructive or destructive interference of all contributions as a function of \Delta\varphi^{\text{\mathcal{M}}}_{\text{01}}. We show results of $\bar{\mathrm{n}}_{\text{1}}$ after t^{\text{\mathcal{M}}}_{\text{01}}\simeq t_{\pi} for $\eta_{\text{0}}\simeq\eta_{\text{1}}\simeq\eta\simeq\{0.9,1.7\}$ in Figs. 3(b) and (c), respectively. For $\eta\simeq 0.9$, transmission is predominantly enabled by the resonance of two distinct channels, corresponding to the carriers and first sidebands at $\text{T}_{\text{0}}$ and $\text{T}_{\text{1}}$, cf. Fig. 3(b, inset). As shown in Fig. 3(b), the measured data is consistent with the Floquet-engineered $\Omega_{\text{AC}}$, considering a linear coupling between harmonic oscillators (dashed line) Bermudez et al. (2012): $\Omega_{\text{AC}}$ and the transfer to $\text{T}_{\text{1}}$ is maximal and robust around \Delta\varphi^{\text{\mathcal{M}}}_{\text{01}}=\pi, while it is significantly suppressed for \Delta\varphi^{\text{\mathcal{M}}}_{\text{01}}=0 and 2$\pi$. A residual coupling for these values can be explained by a residual mismatch of the modulation indices ($\eta_{\text{0}}\neq\eta_{\text{1}}$) and inter-site dephasing. Stronger modulation, see Fig. 3(c,inset) opens additional channels, i.e., leads to larger contribution of upper and lower sidebands. In Figure 3(c) data shows additional features of phase dependent energy transfer, e.g. an additional destructive interference near \Delta\varphi^{\text{\mathcal{M}}}_{\text{01}}=\pi in accordance with the prediction. We note, however, that the two peaks of maximal phonon exchange are narrowed with respect to the idealized theory, and a slight asymmetry appears, which we attribute to anharmonicities resulting from the Coulomb interaction, as well as local trapping potentials.

To demonstrate interference of phonons in two dimensions, akin to the Aharonov-Bohm effect of charged particles under an external magnetic field, all $\text{T}_{\bf j}$ are initialized. We prepare for multi-site coupling by tuning $\omega_{\left\{0,1,2\right\}}$/(2$\pi$) $\simeq$ {4.9, 5.0, 5.1} MHz. We simultaneously apply $\phi^{\mathcal{E}}_{0}(\varphi^{\mathcal{E}}_{0})$ and $\phi^{\mathcal{E}}_{2}(\varphi^{\mathcal{E}}_{2})$, to prepare coherent states with $\bar{\mathrm{n}}_{\text{0}}\simeq$ 5500 and $\bar{\mathrm{n}}_{\text{2}}\simeq$ 5800 phonons and, importantly, a fixed phase relation \Delta\varphi^{\text{\mathcal{E}}}_{\text{02}} 111Note: the phase relation of the ion oscillators results from different constant phase offsets, e.g. caused by supply wiring of the control electrodes, anharmonic contributions of the trapping potential or the duration between the start of excitation and modulation potentials.. Multi-site phonon coupling is activated by applying \phi^{\text{\mathcal{M}}}_{1} with \Omega^{\text{\mathcal{M}}}_{\text{1}}\simeq\Delta\omega_{01}\simeq\Delta\omega_{12} and $\eta_{\text{1}}\simeq 1.8$ during t^{\text{\mathcal{M}}}_{\text{1}}. The lower and upper first sideband at $\text{T}_{1}$ opens transmission channels with the carrier at site $\text{T}_{\text{0}}$ and $\text{T}_{\text{2}}$, respectively, see Fig. 4(b, top). We note that direct phonon exchange between $\text{T}_{\text{0}}$ and $\text{T}_{\text{2}}$  is disabled by the frequency mismatch $\Delta\omega_{02}\gg\Omega_{\text{AC}}$. Results in Fig. 4(b) depict $\bar{\mathrm{n}}_{\text{1}}$  as a function of \Delta\varphi^{\text{\mathcal{E}}}_{\text{02}} for the maximal exchange achieved at t^{\text{\mathcal{M}}}_{\text{1}}=t_{\pi}. While the energy transfer is maximal at \Delta\varphi^{\text{\mathcal{E}}}_{\text{02}}=0 (constructive interference highlighted by red arrow), it is minimal at \Delta\varphi^{\text{\mathcal{E}}}_{\text{02}}=\pi (destructive interference highlighted by gray arrow). In Figure 4(c), we investigate both of the extremal settings in dependence on t^{\text{\mathcal{M}}}_{\text{1}}, depicting the coherent destruction of energy transfer in 2D.

To summarize, we demonstrate Floquet engineering of vibrational excitations in a 2D ion-trap array, present clear signatures of interference effects, and discuss the role of the arising dynamical Peierls phase. In future studies, argon-ion bombardmentHite et al. (2012) or cryogenic environments Labaziewicz et al. (2008) can reduce heating rates by more than two orders of magnitude permitting operation near the motional ground state for durations $\gg 1/\Omega_{\text{AC}}$, as established for short time scales already Kalis (2017). Furthermore triangular lattices, plaquettes to concatenate rhombic ladders and even more complex, non-periodic structures can be realized in future arrays Schaetz et al. (2007, 2013). Combining these techniques with the presented Floquet toolbox may additionally enable to study the interplay of non-linearities, i.e. effective on-site phonon-phonon interaction, with the synthetic gauge fields. This would enable to explore correlated, symmetry-protected topological phases of bosons Huber and Lindner (2011). By exploiting laser cooling and heating mechanisms Lemmer et al. (2018), we can build a phononic analog of photonic lattices Bermudez et al. (2013); Peano et al. (2016); Ozawa et al. (2019); Porras and Fernández-Lorenzo (2019) with a rich interplay between topological and dissipative effects. Application of state-dependent optical potentials, may further extend the quantum-simulation prospects of our platform by enabling state dependent transmission. In particular, by coupling the internal degrees of freedom to the vibrations, one may study bosonic lattice models in the presence of dynamical gauge fields, e.g. famous Aharonov-Bohm physics, cages and edge states Bermudez et al. (2012); Bermudez and Porras (2015).

P.K. and F.H. contributed equally to this work.