⇦Chinese Physics Letters, 2021, Vol. 38, No. 7, Article code 073201Express Letter Rabi Spectroscopy and Sensitivity of a Floquet Engineered Optical Lattice Clock Mo-Juan Yin (尹默娟)1†, Tao Wang (汪涛)2†, Xiao-Tong Lu (卢晓同)1, Ting Li (李婷)1, Ye-Bing Wang (王叶兵)1, Xue-Feng Zhang (张学锋)2*, Wei-Dong Li (李卫东)3*, Augusto Smerzi3,4*, and Hong Chang (常宏)1,5* Affiliations 1Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi'an 710600, China 2Department of Physics, and Center of Quantum Materials and Devices, Chongqing University, Chongqing 401331, China 3Department of Physics and Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China 4QSTAR, INO-CNR, and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy 5School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China Received 8 May 2021; accepted 1 June 2021; published online 8 June 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 61775220, 11804034, 11874094, 12047564, 11874247, 11874246), the Key Research Project of Frontier Science of the Chinese Academy of Sciences (Grant No. QYZDB-SSW-JSC004), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant Nos. XDB21030100 and XDB35010202), the Special Foundation for Theoretical Physics Research Program of China (Grant No. 11647165), the Fundamental Research Funds for the Central Universities (Grant No. 2020CDJQY-Z003), the National Key R&D Program of China (Grant No. 2017YFA0304501), the 111 Project (Grant No. D18001), the Hundred Talent Program of the Shanxi Province (2018), and the EMPIR-USOQS, EMPIR Project co-funded by the European Unions Horizon2020 Research and Innovation Programme and the EMPIR Participating States.
†These authors contributed equally to this work.
*Corresponding authors. Email: zhangxf@cqu.edu.cn; wdli@sxu.edu.cn; augusto.smerzi@ino.it; changhong@ntsc.ac.cn Citation Text: Yin M J, Wang T, Lu X T, Li T, and Wang Y B et al. 2021 Chin. Phys. Lett. 38 073201    Abstract We periodically modulate the lattice trapping potential of a $^{87}$Sr optical clock to Floquet engineer the clock transition. In the context of atomic gases in lattices, Floquet engineering has been used to shape the dispersion and topology of Bloch quasi-energy bands. Differently from these previous works manipulating the external (spatial) quasi-energies, we target the internal atomic degrees of freedom. We shape Floquet spin quasi-energies and measure their resonance profiles with Rabi spectroscopy. We provide the spectroscopic sensitivity of each band by measuring the Fisher information and show that this is not depleted by the Floquet dynamical modulation. The demonstration that the internal degrees of freedom can be selectively engineered by manipulating the external degrees of freedom inaugurates a novel device with potential applications in metrology, sensing and quantum simulations. DOI:10.1088/0256-307X/38/7/073201 © 2021 Chinese Physics Society Article Text The coherent manipulation of quantum systems using periodic modulations, also known as Floquet engineering (FE), is becoming a central paradigm for the realization of synthetic quantum states and Hamiltonians.[1–3] FE has been demonstrated in a variety of different platforms including ultra-cold atoms,[1] photonics[4] and superconducting qubits.[5] In the context of atomic gases trapped in driven optical lattices,[1] the periodic modulation has been employed to renormalize the Hubbard Hamiltonian. In particular, the dynamical modulation can be recast into an effective tunable tunneling.[6] This has led to dynamical control of superfluid-insulator phase transitions,[7] artificial gauge fields[8–10] and topological lattice models.[11–14] In this work, we Floquet engineer an optical lattice clock (OLC) which is among the most accurate precision measurement devices.[15–18] It sets the ground for next standard of time and several proposals aim to exploit their extraordinary stability and accuracy to address fundamental problems ranging from the measurements of physical constants[19] to the detection of gravitational waves.[20,21] OLC consists of an optical local oscillator stabilized by an appropriately chosen two-energy-level transition of atoms trapped in a stationary lattice potential.[22] Usually, a lattice field can significantly modify the transition energies. In OLC, this problem is addressed by engineering the magic-wavelength transition of $^{87}$Sr atoms that is well known to be insensitive to the external trap thanks to a first-order light shift cancelation.[17] Considering the high accuracy and long lifetime of excited clock state, OLC becomes an ideal quantum simulator in several areas, such as the spin orbit coupling and SU(N) Hubbard model.[23,24] However, FE has not been explored so far in OLC platforms. Here we experimentally demonstrate Floquet clock bands created by a dynamical periodic modulation of the incident lattice laser frequency which is different from taking laser power as driving parameter for measuring trap frequency. By trapping the atoms at the magic wave length, we show that the atomic pseudo-spin half [corresponding to the two Zeeman sublevels states $(5s^{2})^{1}\!S_{0} (|g\rangle)$ and $(5s5p)^{3}\!P_{0}(|e\rangle)$ of $F=9/2$] population dynamics is governed by a periodically driven Landau–Zener–Stuckelberg–Majorana Hamiltonian (LZSM).[25,26] We resolve several Floquet quasi-energy bands with ultra-sensitive Rabi spectroscopy. The intensities of each band are independent of driving frequency. The number and the shape of the resonance peaks are controlled by an opportunely tailored multi-frequency driving. We study the spectroscopic sensitivity of each band by measuring the Fisher information, which provides the lower bound in sensitivity in parameter estimation theory.[27,28] We show that the sensitivity is not strongly reduced by the Floquet shaking potential demonstrating that, on the time scales of our experiments, dynamically induced decoherences and thermalization effects can be ignored. Our work opens the design of a new generation of devices for sensing, metrology and quantum simulations where the internal energy structure of an optical lattice clock can be engineered by manipulating the motional atomic degrees of freedom. Experiment Setup. Approximately $10^{4}$ fermionic $^{87}$Sr atoms are cooled down to 3 µK and loaded in a quasi-one-dimensional optical lattice aligned with the $z$ axis. The lattice is created by a counter-propagating laser beam at magic-wavelength $\lambda_{_{\scriptstyle \rm L}} = 813$ nm[29,30] [Fig. 1(a)], so that the atoms at the dipole-forbidden transition energy levels $|g\rangle$ and $|e\rangle$ feel the same lattice potential[17] [Fig. 1(b)]. Meanwhile, the atom is prepared at the Zeeman sublevel of $m_F=+9/2$. We load about one thousand lattice sites separated by barriers of height $V_{0}/E_{r} \approx 90$ ($E_{r}$ is the recoil energy) which hinder inter-site tunneling. We Floquet engineer our system by periodically driving the piezo actuator.[31]
cpl-38-7-073201-fig1.png
Fig. 1. (a) The optical lattice potential is made by a counter-propagating laser ($\lambda_{_{\scriptstyle \rm L}}=813$ nm) along the $z$ direction. The beam waist ($W_0=50\,µ$m) locates at the center of the magneto-optical trap (MOT) and its distance from the high-reflecting mirror is $L\simeq 0.3$ m. The clock laser ($\lambda_{\rm p}=698$ nm) is locked to the ultralow-expansion (ULE) glass cavity and nearly aligned along the same direction of the lattice laser to excite the clock transition. The Floquet modulation (FM) of the lattice laser is controlled with a build-in piezo actuator. (b) At magic-wavelength $\lambda_{_{\scriptstyle \rm L}}$ only one sharp peak at the clock transition is observed in the Rabi spectroscopy (blue curve). (c) When turning on the periodic driving, the carrier peak is split into several Floquet bands as observed in the Rabi spectroscopy (blue curve) and schematically shown in the oval inset.
The frequency of the lattice laser is modulated as $\omega_{_{\scriptstyle \rm L}}(t) = {\bar{\omega}}_{\rm L} + f(t)$, where ${\bar{\omega}}_{\rm L} = 2\pi c/\lambda_{_{\scriptstyle \rm L}}$ is the average lattice carrier frequency and $$ f(t)=\sum_{m=1}^{N} \omega_{m}\sin(\omega_{\rm s} mt)~~ \tag {1} $$ is a $T= 2 \pi/\omega_{\rm s}$ periodic $N$-modes function. We first consider a monochromatic driving $f(t)=\omega_{1}\sin(\omega_{\rm s}t)$, while multi-mode will be discussed later. The intensity of lattice laser along $z$ direction becomes $I=I_{0}\sin^{2}\left(\bar{\omega}_{\rm L}[z+\int v(t) dt]/c\right)$, where $v(t)\simeq \omega_{1}\omega_{\rm s}L \cos(\omega_{\rm s}t)/\bar{\omega}_{\rm L}$ is an effective lattice velocity. In the lattice co-moving frame, the frequency of optical clock laser (CL) $\omega_{\rm p}$ shifts to $\omega_{\rm p}'(t)=[1-v(t)/c]\omega_{\rm p}$ due to the relativistic Doppler effect.[31] Model. When internal clock states are mapped to spin-$1/2$ and described by Pauli matrix $\hat{\sigma}$, the Hamiltonian in the co-moving frame of a single atom interrogated by the clock laser and trapped in a driven periodic potential can be written as $\widehat{H} = {\widehat{H}}_{{\rm ext}} + {\widehat{H}}_{{\rm LZSM}} + {\widehat{H}}_{{\rm c}}$,[31] where $$\begin{align} {\widehat{H}}_{{\rm ext}} ={}& \Big[ \frac{{\widehat{p}}^{2}}{2M} + \frac{\alpha_{0}I_{0}}{2\epsilon_{0}c}e^{- 2r^{2}/W_{0}^{2}}{\sin}^{2}\left( \frac{{\bar{\omega}}_{\rm L}}{c}z \right)\\ & - \frac{M\omega_{1}\omega_{\rm s}^{2}L}{{\bar{\omega}}_{\rm L}}z{\sin}{(\omega}_{\rm s}t)\Big]{\widehat{\sigma}}^{(0)},~~ \tag {2} \end{align} $$ governs the motion of atom, and $$ {\widehat{H}}_{{\rm c}} = \frac{\alpha_{c}I_{0}}{4\epsilon_{0}c}\sin\left( \omega_{\rm s}t \right){{\sin}}^{2}\left( \frac{{\bar{\omega}}_{\rm L}}{c}z \right){\widehat{\sigma}}^{\left( 3 \right)}~~ \tag {3} $$ is coupling Hamiltonian provided by a spin-dependent optical lattice potential (notice that this term has a purely dynamical origin, while spin-dependent periodic potentials have been created with polarized standing waves laser fields before[32,33]). The parameter $\alpha_{c}$ is proportional to the difference between polarizability derivatives of the two spin states calculated at the magic-wavelength.[31] Therefore, the value of coupling constant $\frac{\alpha_{c}I_{0}}{4\epsilon_{0}c} \approx 1$ Hz is so weak that Eq. (3) can be neglected on time scales of the order of a second. This is of the same order of the dephasing time of our optical lattice clock caused by the finite temperature of the atomic sample. The spin dynamics of the atomic gas is governed by the Landau–Zener–Stuckelberg–Majorana Hamiltonian:[26] $$ {\widehat{H}}_{{\rm LZSM}}\left( t \right) = \frac{\hbar}{2}\left( \delta + \omega_{\rm p}\frac{v\left( t \right)}{c} \right){\widehat{\sigma}}_{\boldsymbol{n}}^{\left( 3 \right)} + \frac{g_{\boldsymbol{n}}}{2}{\widehat{\sigma}}_{\boldsymbol{n}}^{\left( 1 \right)},~~ \tag {4} $$ where $\delta = \omega_{0} - \omega_{\rm p}$ is the detuning, $\omega_{0}$ is the clock transition frequency of 87Sr, $g_{\boldsymbol{n}}$ is an effective coupling strength of the atoms with CL. Notice that the spatial driving enters as an effective modulation proportional to $v\left( t \right)$ while the external degrees of freedom of the atoms remain unchanged when ignoring the small linear potential term in Eq. (2): $\hat{\sigma}_{\boldsymbol{n}}$ describes the internal degrees of freedom for atoms at external eigenstates $\boldsymbol{n} = (n_{z},n_{r})$ with the eigenenergies $E_{\boldsymbol{n}}/h = \nu_{z}(n_{z} + {\rm 1/2}) + \nu_{r}(n_{r} + 1)$ of the trapping potential having longitudinal and transverse trap frequencies $\nu_{z} = 64.8$ kHz and $\nu_{r} = 250$ Hz, respectively. Rabi Spectroscopy. The measurements of atomic energies are performed by high-precision clock Rabi spectroscopy that operates at a fractional instability of $10^{-15}$.[34] The spectroscopic CL is locked to an ultralow-expansion cavity having a linewidth of approximately 1 Hz. A slight unavoidable misaligning between CL and lattice axis, see Fig. 1(a), induces a coupling to the suppressed radial motional modes and a correction arising from the thermal distributions of the coupling strength: $$g_{\boldsymbol{n}} = g_{0}e^{- (\eta_{z}^{2} + \eta_{r}^{2})/2}L_{n_{r}}(\eta_{r}^{2})L_{n_{z}}(\eta_{z}^{2}), $$ where $g_{0}/h = 3.3$ Hz, and $L_{n}$ is the $n$th order Laguerre polynomial with Lamb–Dick parameters $\eta_{z} = \sqrt{h/(2M\nu_{z})}/\lambda_{\rm p}$ and $\eta_{r} = \sqrt{h/(2M\nu_{r})}\delta\theta/\lambda_{\rm p}$ with $\delta\theta$ being the misaligned angle between the lattice and the clock laser. At the end of each spectroscopic probe, the number of atoms in the $|g\rangle$ and $|e\rangle$ states are determined using a cycling transition.[31] This provides the normalized population fraction $P_{e}$ of the $|e\rangle$ state. After many repetitions of the measurements, data are collected while the CL is scanned across the clock transition to eventually construct the Rabi spectrum (Fig. 2). Rabi oscillations as a function of the probe pulse time $t_{\rm p}$ are shown in Fig. 3. Both the Rabi spectroscopy and the Rabi oscillations are performed while modulating the system with Floquet periodic driving. Notice that since the measurement processes last only hundreds of milliseconds, we can ignore the spontaneous emission due to the long lifetime of the excited states. According to the Floquet theory,[2] the clock energy levels are split to several Floquet bands (FB) as depicted in Fig. 1(c). The experimental results of the Rabi spectroscopy obtained at different driving modulation amplitudes $\omega_{1}$ and frequencies $\omega_{\rm s}$ are presented in Figs. 2(a)–2(e). We observe, in particular, that (i) sharp FBs are separated by intervals $\omega_{\rm s}$, each band having line width of a few Hz; (ii) the number of FBs is increasing with the renormalized driving amplitude $A = \omega_{1}\omega_{\rm p}L/2{\bar{\omega}}_{\rm L}c$; (iii) the intensity of Floquet Rabi spectra depends on the values of $A$ [not on $\omega_{\rm s}$] after rescaling the detuning as $(\omega_{0} - \omega_{\rm p})/\omega_{\rm s}$. These phenomena can be quantitatively understood within the Floquet theory.[2] Because of the ultra-stable narrow optical CL and $\delta \ll 2\pi\nu_{z}$, the Rabi oscillations mainly occur within a fixed external quantum numbers $\boldsymbol{n}$. We can therefore study the dynamical evolution of the spin populations in an extended Hilbert space, consisting of the direct product of the original spin and Floquet quasi-levels $\lbrack|g\rangle,|e\rangle\rbrack\otimes\left\lbrack 1,e^{\pm i\omega_{\rm s}t},e^{\pm 2i\omega_{\rm s}t},e^{\pm 3i\omega_{\rm s}t},{\rm \ldots } \right\rbrack$. In the resonance region $\omega_{\rm s} \gg g_{n_{z},n_{r}}$,[26] we define an effective Rabi frequency for the $k$th FB as $g_{\boldsymbol{n}}^{k} = g_{\boldsymbol{n}}J_{k}\lbrack2A\rbrack$, where $J_{k}\lbrack2A\rbrack$ is the $k$th order first kind Bessel function. Thus, the excited state population for the $k$th FB is $$ P_{e}^{k}(\delta,t)=\sum_{{\boldsymbol n}}q_{z}(n_z)q_{r}(n_{r})\Big(\frac{g_{{\boldsymbol n}}^{k}}{\hbar R_{k}}\Big)^2 \sin^{2}\Big[\frac{R_{k}}{2} t\Big],~~ \tag {5} $$ where $R_{k} = \sqrt{(g_{\boldsymbol{n}}^{k}/\hbar)^{2} + (\delta - k\omega_{\rm s})^{2}}$. The $q_{z}(n_{z})$ and $q_{r}(n_{r})$ are the statistical distributions of the atoms among eigen-energies of the lattice trapping potential.[35] It is important to notice here that the atomic statistical distributions in periodically driven systems are, in general, not fully understood in the literature. We could still expect a Boltzmann distribution with an effective temperature since, in our case, the Floquet gap $\hbar\omega_{\rm s}/k_{\rm B}$ [$k_{\rm B}$ is the Boltzmann constant] is about a few nK and is much smaller than the longitudinal trap gap energy $h\nu_{z}/k_{\rm B}$, which is several µK. We therefore assume the Boltzmann distributions $q_{z(r)}[n_{z(r)}] = [1 - Z_{z(r)}]Z_{z(r)}^{n_{z(r)}}$ with $Z_{z(r)} = e^{- h\nu_{z(r)}/[k_{\rm B}T_{z(r)}]}$, where the longitudinal and transverse trap effective temperatures $T_{z} = 2.96\,µ$K and $T_{r} = 3.68\,µ$K are extracted by the experimental side bands spectra.[35]
cpl-38-7-073201-fig2.png
Fig. 2. Rabi spectroscopy of the Floquet bands. (a)–(e) Monochromatic driving with different amplitudes $A$. The experimental measurements at the probe time $t_{\rm p}=150$ ms and driving frequencies $\nu_{\rm s}=\omega_{\rm s}/2\pi=50$ Hz (red square) and $\nu_{\rm s}=\omega_{\rm s}/2\pi=100$ Hz (blue diamond) are compared with the theoretical results Eq. (5) (solid black line). (f) Three-frequencies driving with two sets of coefficients: ${\boldsymbol A}_1=\{0.065,0.345,0.243\}$ (mode 1) and ${\boldsymbol A}_2 = \{0.005,0.005,0.16\}$ (mode 2). The measured Rabi spectra of mode 1 (red square) and mode 2 (blue diamond) are presented. In both cases, the probe time is $t_{\rm p}=200$ ms and the driving frequency $\nu_{\rm s}=\omega_{\rm s}/2\pi=50$ Hz. In red ovals, the suppression of second-order FBs when using the Mode-2 set of parameters is shown, demonstrating the possibility of manipulating specific FB. The solid red and black lines are the theoretical Rabi spectra of mode 1 and mode 2, respectively.
The good agreement between the experimental spectra reported in Figs. 2(a)–2(e) and the theoretical predictions of Eq. (5) demonstrates the validity of Boltzmann statistics assumption. Furthermore, from Eq. (5) we gather that the center of Rabi spectrum $P_{e}^{k}(\delta,t_{\rm p})$ is provided by $\omega_{\rm s}$ while the line shapes mainly depend on $A$, in agreement with what observed in (i) and (iii). The point (ii) is the direct consequence of the Bessel functions modulations of the Rabi frequency. By increasing the renormalized driving amplitude $A$, higher order Bessel functions become relevant and an increasing number of FBs emerges. At the same time, the weights of a few of them may decrease or be totally suppressed, as can be observed in the zeroth band at $A = 1.14$ and in the first FB at $A = 1.9$ in Fig. 2. In Figs. 3(a)–3(c) we show the Rabi oscillations at different values of the strength $A$. Notice that at $A = 1.14$, the zeroth band is totally eliminated, as also evident from Fig. 2(c). We can probe the height $P_{e}^{k}(\delta,t_{\rm p})$ of the $k$th FB at a time $t_{\rm p}$ and compare it with the non-driven case at a different time $t_{\rm p}'$, defined as the time when the values of the two peaks are the same. The ratio between Floquet modulated and natural Rabi frequencies is $g_{{\rm eff}}^{k}/g_{0} = t_{\rm p}'/t_{\rm p} = J_{k}\lbrack2A\rbrack$. As shown in Fig. 3(d), this Bessel function dependence emerges quite clearly in the experimental results.
cpl-38-7-073201-fig3.png
Fig. 3. Theoretical (solid lines) and experimental (dashed lines) Rabi oscillations at driving amplitudes (a) $A=0.38$, (b) $A=0.76$ and (c) $A=1.14$. The orange hexagons are for the zeroth band while the blue squares for the first band. (d) The ratio between Floquet modulated Rabi frequencies with Rabi frequencies of the zeroth and first bands measured experimentally are compared with the Bessel function predicted by the Floquet theory.
We now extend the monochromatic driving to $N$ multi modes periodical function, see Eq. (1). In this case the $k$th Floquet level is modulated by a rather complex combination of Bessel functions: $$ \mathcal{J}_{k}[{\boldsymbol A}]=\sum_{\{k_1,k_2,\ldots,k_N\}}\prod_mJ_{k_m}[2A_m],~~ \tag {6} $$ where $\{{\boldsymbol k}_m\}$ is the subset of $\{k_1,k_2,\ldots,k_N\}$ with the constraint $\sum_{m}^{}mk_{m}=k$, and $A_{m} \equiv \omega_{m}\omega_{\rm p}L/2{\bar{\omega}}_{\rm L}c$ is the renormalized driving amplitude for $m$th mode. By appropriately choosing the values of $A_{m}$ of each mode, we independently modulate the FBs while keeping the zeroth band nearly unchanged. As shown in Fig. 2(f), the experimental results of three-frequencies driving are in good agreement with the theoretical prediction of Eq. (5) after replacing $J_{k}\lbrack A\rbrack,$ with $\mathcal{J}_{k}\lbrack\boldsymbol{A}\rbrack$. It would also be possible to create asymmetric distributions by introducing a different phase in each mode. Sensitivity of the Rabi Spectroscopy. We now estimate the spectroscopic sensitivity of the modulated optical clock by measuring the Fisher information (FI). The FI plays a central role in parameter estimation theory since it determines the Cramér–Rao sensitivity lower bound.[21,27,28] It can also be shown that the FI is inversely proportional to the Allan variance,[36] $\sigma_{_{\scriptstyle {\rm Al}}} \sim 1/(\tau F)$, where $\tau$ is the measurement time, whenever time noise correlations of the local oscillator can be neglected. Here the parameter to be estimated is the detuning $\delta$, with the Fisher information $$ F(\delta) = \frac{1}{P_e(\delta)[1-P_e(\delta)]} \Big(\frac{\partial P_e(\delta)}{\partial \delta} \Big)^2.~~ \tag {7} $$
cpl-38-7-073201-fig4.png
Fig. 4. Fisher information of Floquet distributions. (a) The FI as a function of the rescaled detuning with parameter values as in Fig. 2(b) in the cases of zero and finite temperatures with $A=0.76$. (b) FI for the mode-1 and mode-2 three-frequency driving cases, see Fig. 2(f). Experimental (hollow icons) and theoretical (solid icons) values of the maximum Fisher information in the case of (c) $g_0/h=3.3$ Hz and (d) $g_0/h=12$ Hz. The dark filled circles are the maximum theoretical value of the Fisher information, maximized over all possible values of $g_0$, at finite temperature (the maximum Fisher information at zero temperature is $\sim$$9.1 \times 10^{-3}$ at $g_{\rm 0max}/h \simeq 3.34$ Hz while the thermal effects reduces it to $\sim$$5.2 \times 10^{-3}$).
Notice that, since we have dichotomic measurements, the Fisher information coincides with the error propagation expression.[37] Given the symmetry of the probability distributions, the FI of each band has a two-peak structure, with the maximum determined by the competition between a maximum slope and a minimum fluctuation [provided by the denominator of Eq. (7)]. In Figs. 4(a) and 4(b) we show the theoretical values of the Fisher information obtained with $P_{e}(\delta,t)$, Eq. (5), calculated at a fixed $g_{0}$. In Fig. 4(a) we compare the zero and finite temperature cases. It is evident that the value of FI strongly depends on the Floquet band and, as expected, is depleted by the temperature of the atomic gas. In Fig. 4(b) we show the FI for the mode-1 and mode-2 three-frequency drive. In Figs. 4(c) and 4(d) we compare the theoretical values of the FI with the experimental results for different values of $g_{0}$ and $A$. The experimental value of the FI is recovered with a fit of the probability distributions obtained from the Rabi spectra reported in Fig. 2, see Ref. [31]. The agreement between the theoretical calculation and the experimental values is remarkable given that the theoretical Fisher information has been calculated with the ideal probability distributions by only taking in account temperature effects, neglecting all other source of decoherence like the laser fluctuations that are present in the experimental realizations. Furthermore, since in the un-driven case the maximum value of the theoretical FI is approximately $5.2\times10^{-3}$ the sensitivity of lower order Floquet bands is not reduced. Conclusions and Outlook. We have engineered an optical atomic clock by periodically driving the trapping lattice potential, resolved with Rabi spectroscopy several Floquet quasi-energy bands. Also, we have demonstrated the possibility to selectively manipulate a chosen band by appropriately adjusting the driving amplitudes of different driving modes. In future work, it will be possible to shape the inter-well tunneling barriers of the lattice by modulating the potential of Eq. (2) as already demonstrated in Refs. [6–14]. The spin-lattice coupling Hamiltonian Eq. (3) can be switched on to introduce entanglement between spatial and spin coordinates. Furthermore, it will be possible to control the inter-atomic interaction in bosonic clocks with Feshbach resonances.[38] These explorations will open to the possibility of creating a novel generation of quantum simulators with the experimental measure of the Fisher information witnessing multiparticle entanglement.[39,40] Acknowledgment. We thank A. Bertoldi, L. Pezzè and N. Poli for helpful discussions. 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