Floquet Description Research Articles

Phonon excitations of Floquet-driven superfluids in a tilted optical lattice

Tilted lattice potentials with periodic driving play a crucial role in the study of artificial gauge fields and topological phases with ultracold quantum gases. However, driving-induced heating and the growth of phonon modes restrict their use for probing interacting many-body states. Here, we experimentally investigate phonon modes and interaction-driven instabilities of superfluids in the lowest band of a shaken optical lattice. We identify stable and unstable parameter regions and provide a general resonance condition. In contrast to the high-frequency approximation of a Floquet description, we use the micromotion of the superfluids to analyze the growth of phonon modes from slow- to fast-driving frequencies. The model describes phonon excitations in both resonantly and nonresonantly driven systems, with or without a tilted potential. Our observations enable the prediction of stable parameter regimes for quantum-simulation experiments aimed at studying driven systems with strong interactions over extended time scales. Published by the American Physical Society 2024

Pulsed dynamic nuclear polarization: a comprehensive Floquet description.

Dynamic nuclear polarization (DNP) experiments using microwave (mw) pulse sequences are one approach to transfer the larger polarization on the electron spin to nuclear spins of interest. How the result of such experiments depends on the external magnetic field and the excitation power is part of an ongoing debate and of paramount importance for applications that require high chemical-shift resolution. To date numerical simulations using operator-based Floquet theory have been used to predict and explain experimental data. However, such numerical simulations provide only limited insight into parameters relevant for efficient polarization transfer, such as transition amplitudes or resonance offsets. Here we present an alternative method to describe pulsed DNP experiments by using matrix-based Floquet theory. This approach leads to analytical expressions for the transition amplitudes and resonance offsets. We validate the method by comparing computations by these analytical expressions to their numerical counterparts and to experimental results for the XiX, TOP and TPPM DNP sequences. Our results explain the experimental data and are in very good agreement with the numerical simulations. The analytical expressions allow for the discussion of the scaling behaviour of pulsed DNP experiments with respect to the external magnetic field. We find that the transition amplitudes scale inversely with the external magnetic field.

The effect of 1H offset and flip-angle on heteronuclear decoupling efficiency in ROSPAC pulsed sequence: A Floquet description.

A new heteronuclear decoupling sequence for solid-state NMR and magic angle spinning faster than 60kHz was recently introduced [Simion et al., J. Chem. Phys. 157, 014202 (2022)]. It was dubbed ROtor-Synchronized Phase-Alternated Cycles (ROSPAC), and it offers robustness for a large range of chemical shifts and low radio-frequency (RF) powers and is almost independent of the radio-frequency power. Here, we theoretically explore the robustness of the ROSPAC sequence toward 1H offset and RF field inhomogeneities, as well as the spacing effect of the π pulses on the decoupling efficiency. We use a generalized theoretical framework based on the Floquet theory to assess these parameters. The optimum decoupling conditions, where the magnitude of the second-order cross-terms and first-order resonance conditions are small, were identified.

Floquet description of optically pumped magnetometers

We present theoretical description of Voigt and Faraday effect based optically pumped magnetometers using the Floquet expansion. Our analysis describes the spin-operator dynamics of the first, $\hat{F}(t)$, and second, $\hat{F}^2(t)$, order moments and takes into account of different pumping profiles and decoherence effects. We find that the theoretical results are in good agreement with the experimental demonstrations over a wide range of fields and pumping conditions. Finally, the theoretical analysis presented here is generalized and can be extended to different magnetometry schemes with arbitrary pumping profiles and multiple radio-frequency fields.

Thermalization in parametrically driven coupled oscillators

We consider a system of two coupled oscillators one of which is driven parametrically and investigate both classical and quantum dynamics within Floquet description. Characteristic changes in the time evolution of the quantum fluctuations are observed for dynamically stable and unstable regions. Dynamical instability generated by the parametrically driven oscillator leads to infinite temperature thermalization of the undriven oscillator which is evident from the equi-partitioning of energy, reduced density matrix and saturation of entanglement entropy. We also confirm that the classical Lyapunov exponent is correctly captured from the growth rate of ‘unequal time commutator’ of dynamical variables which indicates thermalization stems from the underlying dynamical instability in quantum system.

Improved transfer efficiencies in radio-frequency-driven recoupling solid-state NMR by adiabatic sweep through the dipolar recoupling condition.

The homonuclear radio-frequency driven recoupling (RFDR) experiment is commonly used in solid-state NMR spectroscopy to gain insight into the structure of biological samples due to its ease of implementation, stability towards fluctuations/missetting of radio-frequency (rf) field strength, and in general low rf requirements. A theoretical operator-based Floquet description is presented to appreciate the effect of having a temporal displacement of the π-pulses in the RFDR experiment. From this description, we demonstrate improved transfer efficiency for the RFDR experiment by generating an adiabatic passage through the zero-quantum recoupling condition. We have compared the performances of RFDR and the improved sequence to mediate efficient (13)CO to (13)Cα polarization transfer for uniformly (13)C,(15)N-labeled glycine and for the fibril forming peptide SNNFGAILSS (one-letter amino acid codes) uniformly (13)C,(15)N-labeled at the FGAIL residues. Using numerically optimized sweeps, we get experimental gains of approximately 20% for glycine where numerical simulations predict an improvement of 25% relative to the standard implementation. For the fibril forming peptide, using the same sweep parameters as found for glycine, we have gains in the order of 10%-20% depending on the spectral regions of interest.

Quantification and compensation of the influence of pulse transients on symmetry-based recoupling sequences

Deviations of amplitude and phase of radio-frequency pulses from the desired values, can have a severe impact on the performance of multiple-pulse sequences in NMR spectroscopy. A particular problem are pulse transients that appear every time there is a discontinuity in amplitude or phase. Based on a Floquet description using pulses with arbitrarily shaped amplitudes and phases we present a systematic study of the influence of pulse transients on symmetry-based pulse sequences in solid-state NMR under magic-angle spinning. This treatment explains the dependence of the experimentally observed transfer efficiency on the details of experimental setups. In addition, three approaches are compared which have the aim to re-establish highly efficient recoupling. We demonstrate that the application of transient-compensated pulses as basic elements of symmetry-based sequences leads to a significantly improved robustness of the experiments with respect to variations in the experimental setup.

A Floquet description of phase alternated sequences for efficient homonuclear recoupling in solid perdeuterated systems

A Floquet description of a phase alternated homonuclear recoupling scheme for perdeuterated systems is presented. As a result, we demonstrate improvements in the recoupling efficiency of the DOuble Nucleus Enhanced Recoupling [DONER; J. Am. Chem. Soc. 131 (2009) 17054] technique by utilizing Phase Alternated Recoupling Irradiation Schemes [PARIS; Chem. Phys. Lett. 469 (2009) 342]. The effect of proton and deuterium radio frequency irradiation during recoupling has been systematically studied and theoretical observations have been verified experimentally using a deuterated model compound, l-Alanine, at 10 and 20kHz magic angle spinning frequency. Experimental results are well in agreement with theoretical observations, thereby significantly increasing the recoupling efficiency of conventional DONER in perdeuterated systems.

Radio frequency assisted homonuclear recoupling – A Floquet description of homonuclear recoupling via surrounding heteronuclei in fully protonated to fully deuterated systems

We present a Floquet theory approach for the analysis of homonuclear recoupling assisted by radio frequency (RF) irradiation of surrounding heteronuclear spins. This description covers a broad range of systems from fully protonated to deuterated proteins, focusing in detail on recoupling via protons and deuterons separately as well as simultaneously by the double nucleus enhanced recoupling (DONER) scheme. The theoretical description, supported by numerical simulations and compared to experimental results from a partially deuterated model compound, indicates that in perdeuterated systems setting the RF amplitude equal to the magic angle spinning (MAS) frequency is not necessarily optimal for recoupling via 1H and/or 2H nuclei and modified recoupling conditions are identified.

Understanding two-pulse phase-modulated decoupling in solid-state NMR

A theoretical description of the two-pulse phase-modulated (TPPM) decoupling sequence in magic-angle spinning NMR is presented using a triple-mode Floquet approach. The description is formulated in the radio-frequency interaction-frame representation and is valid over the entire range of possible parameters leading to the well-known results of continuous-wave (cw) decoupling and XiX decoupling in the limit of a phase change of 0 degrees and 180 degrees , respectively. The treatment results in analytical expressions for the heteronuclear residual coupling terms and the homonuclear spin-diffusion terms. It also allows the characterization of all resonance conditions that can contribute in a constructive or a destructive way to the residual linewidth. Some of the important resonance conditions are described for the first time since they are not accessible in previous treatments. The combination of the contributions from the residual couplings and the resonance conditions to the effective Hamiltonian, as obtained in a Floquet description, is shown to be required to describe the decoupling behavior over the full range of parameters. It is shown that for typical spin system and experimental parameters a (13)C linewidth of approximately 12 Hz can be obtained for TPPM decoupling in an organic solid or a protein. This is a major contribution to the experimentally observed linewidths of around 20 Hz and indicates that decoupling techniques are still one of the limiting factors in the achievable linewidths.

Operator-based triple-mode Floquet theory in solid-state NMR

Many solid-state NMR experiments exploit interference effects between time dependencies in the system Hamiltonian to design an effective time-independent Hamiltonian with the desired properties. Effective Hamiltonians can be designed such that they contain only selected parts of the full system Hamiltonian while all other parts are averaged to zero. A general theoretical description of such experiments has to accommodate several time-dependent perturbations with incommensurate frequencies. We describe an extension of the analytical operator-based Floquet description of NMR experiments to situations with three incommensurate frequencies. Experiments with three time dependencies are quite common in solid-state NMR. Examples include experiments which combine magic-angle spinning and radio-frequency irradiation on two nuclei or asynchronous multiple-pulse sequences on a single spin species. The Floquet description is general in the sense that the resulting effective Hamiltonians can be calculated without a detailed knowledge of the spin-system Hamiltonian and can be expressed fully as a function of the Fourier components of the time-dependent Hamiltonian. As a prototype experiment we treat the application of two continuous-wave (cw) radio-frequency fields under magic-angle spinning. Experiments that are included in such a description are Hartmann-Hahn cross polarization or rotary-resonance recoupling experiments with simultaneous cw decoupling.

A physical interpretation of the Floquet description of magic angle spinning nuclear magnetic resonance spectroscopy

A physical interpretation of the Floquet description for magic angle spinning (MAS) nuclear magnetic resonance (NMR) is proposed. The effect of the spatial rotation on the spin system in sample spinning is analysed and described in terms of orbital angular momentum operators. The analogy between rotations in real space and in spin space is emphasized. The transformation properties of the irreducible tensors in real space are used to construct a Floquet Hamiltonian for MAS NMR, that is time independent and comprises one term associated with pure sample rotation. The remaining terms are associated with the spin system, and consist of spin-phonon type Floquet operators generating simultaneous transitions between rotational states and spin states. Finally, two different definitions for the Floquet density operator are compared.

Proton MAS NMR Spectra at High Magnetic Fields and High Spinning Frequencies: Spectral Simulations Using Floquet Theory

Proton magic angle spinning (MAS) spectra of a model spin system, consisting of six protons, were calculated for different values of the external magnetic field and the spinning frequencies. Floquet theory was used to evaluate these spectra. The reduction of the effective homonuclear dipolar interaction for increasing spinning frequency was investigated. The influence of an increase of the external magnetic field and the spinning frequencies on the linewidths of the centerband spectra is discussed. This Floquet description of the rotating proton spin system will assist us in our calculations of the CPMAS spin dynamics of a low abundant spin interacting with a set of coupled protons.

Deuterium Cross-Polarization Magic-Angle Spinning

In this publication a theoretical model is presented that describes cross-polarization magic-angle spinning (CPMAS) NMR experiments on a spin systemS(1)IN, consisting of a set ofNabundant homonuclear spins withI= 12 coupled to a single rare spin withS= 1. The spin evolution during this magic-angle spinning experiment is described using Floquet theory. The model is an extension of the formalism that was recently introduced to describe CPMAS ofS(12)INspin systems. First, experimental results of2H CPMAS experiments on partially deuterated dimethyl sulfone and malonic acid are shown. The rotational-echo intensities of the2H free-induction-decay signals were monitored and plotted as a function of the difference between the intensities of the RF fields applied on the deuterons and the protons during the mixing time. Then the Floquet description of a spin system withS= 1 is presented in order to enable the introduction of the Floquet model for CPMAS NMR. The Floquet Hamiltonian of the rotating quadrupolar spin is defined and the difference between spin locking in the rotating frame and in Floquet space is discussed. This is followed by a description of the spin evolution of theS(1)INsystem during CPMAS experiments. The modified Hartmann–Hahn conditions for these experiments are derived and a methodology for calculating the cross-polarization S-spin signal intensities is demonstrated. The discussion is restricted to spin-1 nuclei with relatively small quadrupole interactions and is directed toward2H CPMAS. S-spin signal intensities as a function of mismatched Hartmann–Hahn conditions are evaluated for powder samples with quadrupolar frequencies of 40 and 120 kHz.

Floquet description of multiphoton processes in Li.

We have made several different types of measurements of the three-photon ionization of Li produced by 3-ps laser pulses and describe the results using a Floquet picture. Over the photon frequency range 15 000 to 15 800 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, Li represents a strongly coupled three-state system with the 2s ground state coupled to the 2p and 3d states by one and two photons, respectively. Energy analysis of the photoelectrons allows the measurement of the intensity dependent shift of the 2s Floquet state during the laser pulse. The shift shows a strong frequency dependence that is not predicted by first-order perturbation theory. We have also measured the total ionization spectrum over several ranges of frequency, as well as the angular distribution of the ionization and the first above-threshold ionization peak for frequencies where the ground state is near resonance with the 4s and 4d excited states. Calculations based on the Floquet Hamiltonian indicate that all of these processes may be understood in terms of a Floquet description.

Adiabatic and diabatic evolution in the multiphoton ionization of sodium.

We have measured the three photon ionization and electron energy spectra of Na from 5-ps laser pulses over the range 16 300 to 17 800 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$. Together these measurements suggest that multiphoton ionization can be understood in terms of a Floquet description. Sharp discontinuities in the ionization probability and Rabi splittings in the electron energy spectra are direct consequences of adiabatic and diabatic evolution, respectively, from the initial zero-field ground state.