The coupling between gauge and matter fields plays an important role in many models of highenergy and condensed matter physics [1][2][3]. In these models, the gauge fields are dynamical quantum degrees of freedom in the sense that they are influenced by the spatial configuration and motion of the matter field. Since the resulting dynamics is hard to compute, it was proposed to implement this fundamental coupling mechanism in quantum simulation platforms with the ultimate goal to emulate lattice gauge theories [4][5][6][7]. So far, synthetic magnetic fields for atoms in optical lattices were intrinsically classical, as these did not feature back-action from the atoms [8,9]. Here, we realize the fundamental ingredient for a density-dependent gauge field by engineering non-trivial Peierls phases that depend on the site occupation of fermions in a Hubbard dimer. Our method relies on breaking timereversal symmetry (TRS) by driving the optical lattice simultaneously at two frequencies. This creates interfering pathways for density-induced tunnelling, each in resonance with the on-site interaction of two fermionic atoms, and controllable in amplitude and phase. We demonstrate a technique to quantify the amplitude of the resulting density-assisted tunnelling matrix element and to directly measure its Peierls phase with respect to the single-particle hopping. The tunnel coupling features two distinct regimes as a function of the two modulation amplitudes, which can be characterised by a Z 2 -invariant. Moreover, we provide a full tomography of the winding structure of the Peierls phase around a Dirac point that appears in the driving parameter space. For future experiments, this structure provides unique tunability of the associated density-dependent gauge field by using modulation parameters with temporal or spatial dependencies.The fundamental manifestation of a gauge field in electromagnetism is the Lorentz force acting on charged particles. In ultracold Bose and Fermi gases, the charge neutrality of the atoms requires to engineer synthetic magnetic fields. This has been achieved for bulk systems by a rotation of the gas or a suitable coupling of momentum states via Raman lasers [10,11]. For a tight-binding model on a lattice, the equivalent of an Aharonov-Bohm phase can be synthesized with Peierls phases resulting from a complex-valued tunnelling matrix element. Such phases can be engineered in a Floquet approach by a suitable driving scheme [12,13], which has been used in cold atom experiments to generate static gauge fields [14][15][16][17]. In contrast to these classical fields, the simulation of dynamical gauge fields requires the implementation of a back-action mechanism that couples the gauge and matter fields. One possibility is to engineer densitydependent gauge fields by making use of interactions [18]. Such a scheme has recently been implemented experimentally by adding a directional mean-field shift in momentum space to a Bose-Einstein condensate [19]. For tight-binding models a back-action mechanism encode...

The recent maturation of hybrid quantum devices has led to significant enhancements in the functionality of a wide variety of quantum systems. In particular, harnessing mechanical resonators for manipulation and control has expanded the use of two-level systems in quantum information science and quantum sensing. In this letter, we report on a monolithic hybrid quantum device in which strain fields associated with resonant vibrations of a diamond cantilever dynamically control the optical transitions of a single nitrogen-vacancy (NV) defect center in diamond. We quantitatively characterize the strain coupling to the orbital states of the NV center, and with mechanical driving, we observe NV-strain couplings exceeding 10 GHz.Furthermore, we use this strain-mediated coupling to match the frequency and polarization dependence of the zero-phonon lines of two spatially separated and initially distinguishable NV centers. The experiments demonstrated here mark an important step toward engineering a quantum device capable of realizing and probing the dynamics of non-classical states of mechanical resonators, spin-systems, and photons.2

We study the dynamics and timescales of a periodically driven Fermi-Hubbard model in a threedimensional hexagonal lattice. The evolution of the Floquet many-body state is analyzed by comparing it to an equivalent implementation in undriven systems. The dynamics of double occupancies for the near-and off-resonant driving regime indicate that the effective Hamiltonian picture is valid for several orders of magnitude in modulation time. Furthermore, we show that driving a hexagonal lattice compared to a simple cubic lattice allows to modulate the system up to 1 s, corresponding to hundreds of tunneling times, with only minor atom loss. Here, driving at a frequency close to the interaction energy does not introduce resonant features to the atom loss.Floquet engineering is a versatile method to implement novel, effectively static Hamiltonians by applying a periodic drive to a quantum system [1][2][3]. For long timescales, a limitation for this method to create interesting many-body states is eventually the heating to an infinite temperature, caused by the presence of integrability breaking terms such as interactions [4,5]. For very short time scales, an obvious limit is set by the duration of a single cycle, which cannot be captured by a static Hamiltonian. In general, the launch of the drive causes complex dynamics on different timescales in a many-body system [6][7][8]. Theoretical considerations suggest that an effective Hamiltonian picture can still remain valid for some intermediate timescale required to create many-body phases [7,[9][10][11][12][13][14][15][16]. Developing an experimental approach to identify relevant timescales in a periodically driven quantum system with interactions is thus a timely challenge.In this Letter, we investigate the Floquet dynamics of a periodically driven Fermi-Hubbard model, which is realized with interacting fermions in a three-dimensional optical lattice. Our approach allows us to experimentally compare the evolution of an observable in a driven system with the equivalent dynamics in an undriven Hamiltonian. The evolution of the entire many-body state is complex (see Fig. 1a) -while local processes, like the tunneling, play a role on short timescales, the trapping potential sets a timescale for global thermalization. In addition, deviations to the expected behavior in the effective Hamiltonian might arise for very long modulation times. In the comparison, we analyze this evolution of the many-body state due to a change of (effective) Hubbard parameters and disentangle it from heating in driven systems which cannot be captured by an effective static model. The latter can be understood as unwanted absorption processes, which in the presence of interactions may be resonant at any driving frequency, since the energy spectrum becomes continuous [17][18][19][20]. Although resonant processes can be desired to realize a specific Floquet Hamiltonian [21][22][23][24][25][26][27][28][29][30][31][32][33][34], a general understanding of the dynamics of strongly correlated driven quantum st...

We study the dynamics of double occupations in a strongly interacting, periodically driven Fermi-Hubbard model using ultracold, fermionic atoms in an optical lattice and the nonequilibrium extension of dynamical mean field theory. Ramping up the amplitude and varying the frequency of the drive we demonstrate the validity of the effective static Hamiltonian description in the far offresonant regime and its breakdown due to energy absorption closer to the resonance. In the case of near-resonant driving, we investigate the effect of the driving amplitude and detuning as well as the adiabaticity of the switching into a state with effectively reduced interaction. A good agreement between experiment and theory is found in both regimes, which establishes a high level of control in the quantum simulations and the predictive power of the numerical approach.Quantum simulation exploits the high degree of control over a quantum system, such as ultracold atoms, to study many-body effects which are difficult to predict analytically or numerically [1][2][3][4]. One of the target systems in the condensed matter context is the Fermi-Hubbard model, which captures essential effects of electronic correlations in solids. For equilibrium properties, cold atom simulators and powerful numerical techniques provide access to temperature regimes, where nontrivial quantum correlations start to emerge. These methods have been successfully benchmarked against each other and used to reveal the physics of the model [5][6][7][8][9][10][11]. Exposing such correlated lattice models to a periodic driving force creates a broad range of interesting effects [12,13]. Applications of this so-called Floquet engineering concept range from the coherent control of high-temperature superconductors by intense laser fields [14,15] to the realization of topological phases in cold atomic systems [13,16]. The nonequilibrium nature of these effects however pushes quantum simulators and numerical methods to their limits, where a direct comparison demands exceptionally high control over experimental parameters and a sufficient accuracy of the numerical methods. Here, we address this challenge of an experiment-theory comparison for the driven Fermi-Hubbard model on a three-dimensional lattice.The properties of periodically driven systems are often described in terms of an effective Floquet Hamiltonian, which is static and can be derived from high-frequency expansions or time-dependent Schrieffer-Wolff transformations [17][18][19][20]. However, these expansions are only useful if there exists a hierarchy of energy scales which makes it possible to neglect higher order terms. Furthermore, a realistic description of a Floquet-engineered state must take into account the possibly non-thermal energy distribution in the system as well as the switch-on procedure and the continuous energy absorption from the periodic drive [18,[21][22][23][24]. In experiments, unwanted excitation processes are induced by the driving [25][26][27][28][29][30] or by fluctu-ations of static param...

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