The extended quantum Rabi models make a significant contribution to understand the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, anisotropic Rabi model and asymmetric Rabi model, into rotating frame, and regard them as periodically driven quantum systems. The analytical solutions of the quasi-energy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the non-stroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, atom-field correlation, etc. For anisotropic Rabi model, the quasi energy fits well with the numerical results even when the rotating-wave coupling is in the deep-strong coupling regime $g\simeq\hbarω$ if the counterrotating terms is small enough compared to the driving frequency. Avoided level crossing may occur for quasi energy with the same parity when the positive branch spectrum lines for the total excitation number $N$ cross the negative branch lines for $N+2$, while the high frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical study of the long-time evolution of population and figure out the analytical method is credible for the population dynamics. For asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.
Quantum metrology with ultra-high precision usually requires atoms prepared in an ultra-stable environment with well-defined quantum states. Thus, in optical lattice clock systems deep lattice potentials are used to trap ultra-cold atoms. However, decoherence, induced by Raman scattering and higher order light shifts, can significantly be reduced if atomic clocks are realized in shallow optical lattices. On the other hand, in such lattices, tunneling among different sites can cause additional dephasing and strongly broadening of the Rabi spectrum. Here, in our experiment, we periodically drive…
Low-frequency fiber-optic vibration sensing with a Floquet-engineered optical lattice clock
National Time Service Center, Chinese Academy of Sciences（中国科学院国家授时中心）；School of Physics, Northwest University（西北大学物理学院）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文与空间科学学院）
We reconsider the periodically driven two level system and especially the Rabi problem with linear polarization. The Floquet theory of this problem can be reduced to its classical limit, i.e., to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to J.H. Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approxim…
Floquet engineering of isolated systems is often based on the concept of the effective time-independent Floquet Hamiltonian, which describes the stroboscopic evolution of a periodically driven quantum system in steps of the driving period and which is routinely obtained analytically using high-frequency expansions. The generalization of these concepts to open quantum systems described by a Markovian master equation of Lindblad type turns out to be non-trivial: On the one hand, already for a two-level system two different phases can be distinguished, where the effective time-independent Floquet…
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Center for Quantum Science and Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院量子科学与工程中心）；Dipartimento di Fisica “A. Volta”, Università di Pavia（帕维亚大学“A.沃尔塔”物理学系）
We consider the solution of the equation of motion of a classical/quantum spin subject to a monochromatical, elliptically polarized external field. The classical Rabi problem can be reduced to third order differential equations with polynomial coefficients and hence solved in terms of power series in close analogy to the confluent Heun equation occurring for linear polarization. Application of Floquet theory yields the physically interesting quasienergy as a function of the parameters of the problem and expressions for the Bloch-Siegert shift of resonance frequencies. Various limit cases cases…
We develop a low-frequency perturbation theory in the extended Floquet Hilbert space of a periodically driven quantum systems, which puts the high- and low-frequency approximations to the Floquet theory on the same footing. It captures adiabatic perturbation theories recently discussed in the literature as well as diabatic deviation due to Floquet resonances. For illustration, we apply our Floquet perturbation theory to a driven two-level system as in the Schwinger-Rabi and the Landau-Zener-Stückelberg-Majorana models. We reproduce some known expressions for transition probabilities in a simpl…
We analyze the convergence properties of the connected moments expansion (CMX) for the Rabi Hamiltonian. To this end we calculate the moments and connected moments of the Hamiltonian operator to a sufficiently large order. Our large--order results suggest that the CMX is not reliable for most practical purposes because the expansion exhibits considerable oscillations.
We theoretically calculate and experimentally measure the beyond-mean-field (BMF) equation of state in a coherently-coupled two-component Bose-Einstein condensate (BEC) in the regime where averaging of the interspecies and intraspecies coupling constants over the hyperfine composition of the single-particle dressed state predicts the exact cancellation of the two-body interaction. We show that with increasing the Rabi-coupling frequency $Ω$, the BMF energy density crosses over from the nonanalytic Lee-Huang-Yang (LHY) scaling $\propto n^{5/2}$ to an expansion in integer powers of density, wher…
We theoretically investigate basic properties of nonequilibrium steady states of periodically-driven open quantum systems based on the full solution of the Maxwell-Bloch equation. In a resonantly driving condition, we find that the transverse relaxation, also known as decoherence, significantly destructs the formation of Floquet states while the longitudinal relaxation does not directly affect it. Furthermore, by evaluating the quasienergy spectrum of the nonequilibrium steady states, we demonstrate that the Rabi splitting can be observed as long as the decoherence time is as short as one thir…
The Floquet eigenvalue problem is analyzed for periodically driven Friedrichs models on discrete and continuous space. In the high-frequency regime, there exists a Floquet bound state consistent with the Floquet-Magnus expansion in the discrete Friedrichs model, while it is not the case in the continuous model. In the latter case, however, the bound state predicted by the Floquet-Magnus expansion appears as a metastable state whose lifetime diverges in the limit of large frequencies. We obtain the lifetime by evaluating the imaginary part of the quasi-energy of the Floquet resonant state. In t…
We explore the long-time dynamics of Rabi model in a driven-dissipative setting and show that, as the atom-cavity coupling strength becomes larger than the cavity frequency, a new time scale emerges. This time scale, much larger than the natural relaxation time of the atom and the cavity, leads to long-lived metastable states susceptible to being observed experimentally. By applying a Floquet-Liouville approach to the time-dependent master equation, we systematically investigate the set of possible metastable states. We find that the properties of the metastable states can differ drastically f…
We present a frequency-renormalized multipolaron expansion method to explore the ground state of quantum Rabi model (QRM). The main idea is to take polaron as starting point to expand the ground state of QRM. The polarons are deformed and displaced oscillator states with variationally determined frequency-renormalization and displacement parameters. This method is an extension of the previously proposed polaron concept and the coherent state expansion used in the literature, which shows high efficiency in describing the physics of the QRM. The proposed method is expected to be useful for solvi…
We demonstrate the emergence of a non-equilibrium superradiant phase in the dissipative Rabi-Dicke model. This phase is characterized by a photonic steady state that oscillates with a frequency close to the cavity frequency, in contrast to the constant photonic steady state of the equilibrium superradiant phase in the Dicke model. We relate this superradiant phase to the population inversion of Floquet states by introducing a Schwinger representation of the driven two-level systems in the cavity. This inversion is depleted near Floquet energies that are resonant with the cavity frequency to su…
We present a physically motivated variational wave function for the ground state of the asymmetric quantum Rabi model (AQRM). The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states, and relies only on three variational parameters in the regimes of interest where the squeezing effect becomes negligible. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. We use the variational result to calculate various relevant physical observables of the ground state…
Floquet engineering offers a compelling approach for designing the time evolution of periodically driven systems. We implement a periodic atom-light coupling to realize Floquet atom optics on the strontium ${}^1\!S_0\,\text{-}\, {}^3\!P_1$ transition. These atom optics reach pulse efficiencies above $99.4\%$ over a wide range of frequency offsets between light and atomic resonance, even under strong driving where this detuning is on the order of the Rabi frequency. Moreover, we use Floquet atom optics to compensate for differential Doppler shifts in large momentum transfer atom interferometers…
We consider the dynamics of a two-level system (qubit) driven by strong and short resonant pulses in the framework of Floquet theory. First we derive analytical expressions for the quasienergies and Floquet states of the driven system. If the pulse amplitude varies very slowly, the system adiabatically follows the instantaneous Floquet states, which acquire dynamical phases that depend on the evolution of the quasienergies over time. The difference between the phases acquired by the two Floquet states corresponds to a qubit state rotation, generalizing the notion of Rabi oscillations to the ca…
A polemic arose recently about the applicability of the $t$-expansion method to the calculation of the ground state energy $E_0$ of the Rabi model. For specific choices of the trial function and very large number of involved connected moments, the $t$-expansion results are rather poor and exhibit considerable oscillations. In this letter, we formulate the $t$-expansion method for trial functions containing two free parameters which capture two exactly solvable limits of the Rabi Hamiltonian. At each order of the $t$-series, $E_0$ is assumed to be stationary with respect to the free parameters.…
The Floquet-Magnus expansion is a useful tool to calculate an effective Hamiltonian for periodically driven systems. In this study, we investigate the convergence of the expansion for a one-body nonlinear system in a continuous space, a driven anharmonic oscillator. In this model, all eigenstates of the time evolution operator are found to be localized in energy space, and the expectation value of the energy is bounded from above. We first propose a general procedure to estimate the radius of convergence of the Floquet-Magnus expansion for periodically driven systems with an unbounded energy s…
We study numerically the onset of higher-level excitations and resonance frequency shifts in the generalized multilevel Rabi model with dispersive coupling under strong driving. The response to a weak probe is calculated using the Floquet method, which allows us to calculate the probe spectrum and extract the resonance frequency. We test our predictions using a superconducting circuit consisting of transmon coupled capacitively to a coplanar waveguide resonator. This system is monitored by a weak probe field, and at the same time driven at various powers by a stronger microwave tone. We show t…
We apply the Picard and Magnus expansions to both the semiclassical and the quantum Rabi model, with a switchable matter-field coupling. The case of the quantum Rabi model ia a paradigmatic example of finite-time quantum electrodynamics (QED), and in this case we build an intuitive diagrammatic representation of the Picard series. In particular, we show that regular oscillations in the mean number of photons, ascribed to the dynamical Casimir effect (DCE) for the the generation of photons and to the anti-DCE for their destruction, take place at twice the resonator frequency $ω$. Such oscillati…
We study the dynamical properties of the quantum Rabi model within a systematic expansion method. Based on the observation that the parity symmetry of the Rabi model is kept during the evolution of the states, we decompose the initial state and the time-dependent one into a part of a positive and a negative parity expanded by the superposition of the coherent states. The evolutions for the corresponding positive and the negative parity are obtained, where the expansion coefficients in the dynamical equations are known from the recurrence relation derived.
The quantum system under periodical modulation is the simplest path to understand the quantum non-equilibrium system, because it can be well described by the effective static Floquet Hamiltonian. Under the stroboscopic measurement, the initial phase is usually irrelevant. However, if two uncorrelated parameters are modulated, their relative phase can not be gauged out, so that the physics can be dramatically changed. Here, we simultaneously modulate the frequency of the lattice laser and the Rabi frequency in an optical lattice clock (OLC) system. Thanks to ultra-high precision and ultra-stabi…
We study the population trapping extensively in a periodically driven Rydberg pair. The periodic modulation of the atom-light detuning effectively suppresses the Rabi couplings and, together with Rydberg-Rydberg interactions, leads to the state-dependent population trapping. We identify a simple yet a general scheme to determine population trapping regions using driving induced resonances, the Floquet spectrum, and the inverse participation ratio. Contrary to the single atom case, we show that the population trapping in the two-atom setup may not necessarily be associated with level crossings …
We consider a spin $s$ subjected to both a static and an orthogonally applied oscillating, circularly polarized magnetic field while being coupled to a heat bath, and analytically determine the quasi\-stationary distribution of its Floquet-state occupation probabilities for arbitrarily strong driving. This distribution is shown to be Boltzmannian with a quasitemperature which is different from the temperature of the bath, and independent of the spin quantum number. We discover a remarkable formal analogy between the quasithermal magnetism of the nonequilibrium steady state of a driven ideal pa…
Floquet-Magnus (FM) expansion theory is a powerful tool in periodically driven (Floquet) systems under high-frequency drives. In closed systems, it dictates that their stroboscopic dynamics under a time-periodic Hamiltonian is well captured by the FM expansion, which gives a static effective Hamiltonian. On the other hand, in dissipative systems driven by a time-periodic Liouvillian, it remains an important and nontrivial problem whether the FM expansion gives a static Liouvillian describing continuous-time Markovian dynamics, which we refer to as the Liouvillianity of the FM expansion. We ans…
We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems. Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory. The final results are equivalent to those obtained within a different approach [Phys.\ Rev.\ A {\bf 68}, 013820 (2003), Phys.\ Rev.\ X {\bf 4}, 031027 (2014)] and can also be related to the Floquet-Magnus expansion [J.\ Phys.\ A {\bf 34}, 3379 (2000)]. We discuss that the dependence on the d…
Much recent experimental effort has focused on the realization of exotic quantum states and dynamics predicted to occur in periodically driven systems. But how robust are the sought-after features, such as Floquet topological surface states, against unavoidable imperfections in the periodic driving? In this work, we address this question in a broader context and study the dynamics of quantum systems subject to noise with periodically recurring statistics. We show that the stroboscopic time evolution of such systems is described by a noise-averaged Floquet superoperator. The eigenvectors and -v…
Three-body correlations, which arise between spin-polarized electrons in the first excited Landau level, are believed to play a key role in the emergence of enigmatic non-Abelian fractional quantum Hall (FQH) effects. Inspired by recent advances in Floquet engineering, we investigate periodic driving of anisotropic two-body interactions as a route for controllably creating and tuning effective three-body interactions in the FQH regime. We develop an analytic formalism to describe this Floquet-FQH protocol, which is distinct from previous approaches that instead focus on bandstructure engineeri…
The Floquet-Magnus expansion of unbounded operators
Institute of Theoretical Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学理论物理研究所）；School of Mathematical Sciences, Lancaster University（兰卡斯特大学数学科学学院）
We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combing Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equa…
RING: Rabi oscillations induced by nonresonant geometric drive
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN-BME-BCE Quantum Technology Research Group, Budapest University of Technology and Economics（匈牙利研究与教育网络-布达佩斯技术与经济大学-布达佩斯经济与商业学院量子技术研究组）
Floquet framework for driven polar quantum systems
Institute of Theoretical Physics and Astronomy, Vilnius University（维尔纽斯大学理论物理与天文学研究所）；Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń（托伦尼古拉·哥白尼大学物理、天文与信息学院物理研究所）；Institute of Advanced Studies, Nicolaus Copernicus University in Toruń（托伦尼古拉·哥白尼大学高级研究所）
A Variational Framework for Time-Dependent Quantum Systems with Applications to Floquet Hamiltonians
Memorial University of Newfoundland and Labrador（纽芬兰与拉布拉多纪念大学）；National Research Council Canada（加拿大国家研究委员会）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）；Compute Everything Technologies Ltd.（Compute Everything科技有限公司）
Nonequilibrium steady states (NESSs) in periodically driven dissipative quantum systems are vital in Floquet engineering. We develop a general theory for high-frequency drives with Lindblad-type dissipation to characterize and analyze NESSs based on the high-frequency (HF) expansion with linear algebraic numerics and without numerically solving the time evolution. This theory shows that NESSs can deviate from the Floquet-Gibbs state depending on the dissipation type. We show the validity and usefulness of the HF-expansion approach in concrete models for a diamond nitrogen-vacancy (NV) center, …
We investigate both pure and mixed states Floquet dynamical quantum phase transition (DQPT) in the periodically time-dependent extended XY model. We exactly show that the proposed Floquet Hamiltonian of interacting spins can be expressed as a sum of noninteracting quasi-spins imposed by an effective time dependent magnetic field (Schwinger-Rabi model). The calculated Chern number indicates that there is a topological transition from nonadiabatic to adiabatic regime. In the adiabatic regime, the quasi-spins trace the time dependent effective magnetic field and then oscillate between spin up and…
We develop a systematic variational coherent-squeezed-state expansion for the ground state of the quantum Rabi model, which includes an additional squeezing effect with comparisons to previous coherent-state approach. For finite large ratio between the atomic and field frequency, the essential feature of the ground-state wave function in the super-radiant phase appears, which has a structure of two delocalized wake packets. The single-peaked wave function with one coherent-squeezed state works well even around the critical regime, exhibiting the advantage over the coherent-state method. As the…
We determine the relevant spinor valued wave functions for a two-dimensional quantum ring in the presence of Rashba-type spin-orbit interaction (SOI). The case of constant SOI strength is considered first, then we investigate the physical consequences of time-dependent (oscillating) SOI strength. Floquet's method is applied to find time-dependent eigenspinors, and it is shown that the Floquet quasienergies and thus their differences (the generalized Rabi frequencies) are determined by the radial boundary conditions. Time evolution of various initial states is calculated.
We study heating dynamics in isolated quantum many-body systems driven periodically at high frequency and large amplitude. Combining the high-frequency expansion for the Floquet Hamiltonian with Fermi's golden rule (FGR), we develop a master equation termed the Floquet FGR. Unlike the conventional one, the Floquet FGR correctly describes heating dynamics, including the prethermalization regime, even for strong drives, under which the Floquet Hamiltonian is significantly dressed, and nontrivial Floquet engineering is present. The Floquet FGR depends on system size only weakly, enabling us to an…
Trimer Dynamics in Floquet-driven arrays of Rydberg Atoms
Università degli Studi di Padova（帕多瓦大学）；INFN Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；Padua Quantum Technologies Research Center (QTECH)（帕多瓦量子技术中心）
We study electric dipole spin resonance caused by sub-terahertz (THz) radiation in a multilevel finite-size quantum dot formed in a nanowire focusing on the range of driving electric fields amplitudes where a strong interplay between the Rabi spin oscillations and tunneling from the dot to continuum states can occur. A strong effect of the tunneling on the spin evolution in this regime occurs due to formation of mixed spin states. As a result, the tunneling strongly limits possible spin manipulations time. We demonstrate a backaction of the spin dynamics on the tunneling and position of the el…
Recent experimental advances enable the manipulation of quantum matter by exploiting the quantum nature of light. However, paradigmatic exactly solvable models, such as the Dicke, Rabi or Jaynes-Cummings models for quantum-optical systems, are scarce in the corresponding solid-state, quantum materials context. Focusing on the long-wavelength limit for the light, here, we provide such an exactly solvable model given by a tight-binding chain coupled to a single cavity mode via a quantized version of the Peierls substitution. We show that perturbative expansions in the light-matter coupling have …
We use Floquet theory and the High-Frequency expansion to derive an effective Hamiltonian for electrons coupled to an off resonant cavity mode, either in its vacuum or driven by classical light. For vacuum fields, we show that long-range hopping and cavity-mediated interactions arise as a direct consequence of quantum fluctuations. As an application, this method is applied to the Su-Schrieffer-Heeger (SSH) model. At high light-matter coupling, our results reveal significant deviations from mean-field predictions, with our framework capturing light-matter entanglement through the Floquet microm…
The asymmetric quantum Rabi model with broken parity invariance shows spectral degeneracies in the integer case, that is when the asymmetry parameter equals an integer multiple of half the oscillator frequency, thus hinting at a hidden symmetry and accompanying integrability of the model. We study the expectation values of spin observables for each eigenstate and observe characteristic differences between the integer and noninteger cases for the asymptotics in the deep strong coupling regime, which can be understood from a perturbative expansion in the qubit splitting. We also construct a pare…
