Transient phenomena of phase modulated cut-off wavepackets are explored by deriving an exact general solution to Schrödinger's equation for finite range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual \textit{self-induced two-level system} with energies $E_{+}$, and $E_{-}$, due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi-oscillations characterized by a frequency $Ω=(E_{+}-E_{-})/\hbar$, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a \textit{quantum beat} effect with a beating frequency, $Ω$. We also find a regime characterized by a \textit{time-diffraction} phenomenon that allows to measure unambiguously the delay-time, which can be described by an exact analytical formula. It is found that the delay-time agrees with the phase-time only for the case of strictly monochromatic waves.
We study the coherent dynamics of an excited two-level atom in a vacuum optomechanical cavity and find that the original atom-cavity Rabi oscillation is sinusoidally modulated by the light-mechanics coupling as the Rabi splitting is on resonance with the mechanical mode. We develop an analytic model in a three-dimensional Hilbert subspace to explain this phenomenon and employ numerical simulations of the density-matrix master equation to confirm our analysis. We also show that the modulated Rabi oscillation survives in presence of dissipations and other non-ideal factors.
Two proof-of-principle experiments towards T1-limited magnetic resonance imaging with NV centers in diamond are demonstrated. First, a large number of Rabi oscillations is measured and it is demonstrated that the hyperfine interaction due to the NV's 14N can be extracted from the beating oscillations. Second, the Rabi beats under V-type microwave excitation of the three hyperfine manifolds is studied experimentally and described theoretically.
We study the coherent dynamics of a qubit excited by an amplitude-modulated electromagnetic field under the Rabi resonance when the frequency of the low-frequency modulation field matches the Rabi frequency in the high-frequency field. Due to destructive interference of multiple photon processes at the ultrastrong coupling between the qubit and the low-frequency driving field, Rabi oscillations result exclusively from the Bloch-Siegert effect. It is directly observed in the time-resolved coherent dynamics as the Bloch-Siegert oscillation. In this case, triplets in Fourier spectra of the cohere…
We investigate the bias-modulated dynamics of a strongly driven two-level system using the counter-rotating-hybridized rotating-wave (CHRW) method. This CHRW method treats the driving field and the bias on equal footing by a unitary transformation with two parameters $ξ$ and $ζ$, and is nonperturbative in driving strength, tunneling amplitude or bias. In addition, this CHRW method is beyond the traditional rotating-wave approximation (Rabi-RWA) and yet by properly choosing the two parameters $ξ$ and $ζ$, the transformed Hamiltonian takes the RWA form with a renormalized energy splitting and a …
We study a non-Hermitian version of the Rabi model, where a two-level system is periodically driven with an imaginary-valued drive strength, leading to alternating gain and loss. In the Floquet picture, the model exhibits PT symmetry, which can be broken when the drive is sufficiently strong. We derive the boundaries of the PT phase diagram for the different resonances by doing perturbation theory beyond the rotating-wave approximation. For the main resonance, we show that the non-Hermitian analog of the Bloch-Siegert shift corresponds to maximal PT-breaking. For the higher-order resonances, w…
We studied the interaction of a two-level atom with a frequency modulated cavity mode in an ideal optical cavity. The system, described by a Jaynes-Cumming Hamiltonian, gave rise to a set of stiff nonlinear first order equations solved numerically using implicit and semi-implicit numerical algorithms. We explored the evolution of the atomic system using nonlinear dynamics tools, like time series, phase plane, power spectral density, and Poincaré sections plots, for monochromatic and bichromatic modulations of the cavity field. The system showed quasiperiodic and possibly chaotic behavior when …
Anisotropic quantum Rabi model is a generalization of quantum Rabi model, which allows its rotating and counter-rotating terms to have two different coupling constants. It provides us with a fundamental model to understand various physical features concerning quantum optics, solid-state physics, and mesoscopic physics. In this paper, we propose an experimental feasible scheme to implement anisotropic quantum Rabi model in a circuit quantum electrodynamics system via periodic frequency modulation. An effective Hamiltonian describing the tunable anisotropic quantum Rabi model can be derived from…
Atomic spectroscopy, an essential tool for frequency estimation, is widely used in quantum sensing. Atom-atom interaction can be used to generate entanglement for achieving quantum enhanced sensing. However, atom-atom interaction always induces collision shift, which brings systematic error in determining the resonance frequency. Contradiction between utilizing atom-atom interaction and suppressing collision shift generally exists in atomic spectroscopy. Here, we propose an antisymmetric Rabi spectroscopy protocol without collision shift in the presence of atom-atom interactions. We analytical…
Rabi nutations of a single electron spin in a single defect center have been detected. The coherent evolution of the spin quantum state is followed via optical detection of the spin state. Coherence times up to several microseconds at room temperature have been measured. Optical readout of the spin states leads to decoherence, equivalent to the so-called Zeno effect. Quantum beats between electron spin transitions in a single spin Hahn echo experiment are observed. A closer analysis reveals that beats also result from the hyperfine coupling of the electron spin to a single 14N nuclear spin. Th…
We study the dynamics of the Rabi Hamiltonian in the medium coupling regime with $\left\vert g/ω\right\vert \sim 0.07$, where $g$ is atom-field coupling constant, $ω$ is the field frequency, for the quantum state with average photon number $\bar{n}\sim 10^{4}$. We map the original Hamiltonian to an effective one, which describes a tight-binding chain subjected to a staggered linear potential. It is shown that the photon probability distribution of a Gaussian-type state exhibits the amplitude modulated Bloch oscillation (BO), which is a superposition of two conventional BOs with a half-BO-perio…
We study Rabi oscillations of the second-order Raman transition realized on dressed states of a qubit excited by an amplitude-modulated microwave field. The co-rotating component of the ultrastrong low-frequency modulation field excites virtual multiple photon processes between the dressed states and forms the Rabi frequency in the so-called rotating wave approximation (RWA). The counter-rotating modulation component also gives a significant contribution to the Rabi frequency owing to the Bloch--Siegert effect. It is shown that for properly chosen parameters of the modulation field and qubit, …
We analyze the strong-coupling dynamics of a driven harmonic oscillator whose energy is modulated by a continuum of other bosonic modes. This type of system-bath interaction appears, for example, in optomechanical or equivalent circuit QED setups, where the frequency of a confined photonic mode depends linearly on a fluctuating boundary. Compared to the canonical spin-boson model, where coupling to bath modes only leads to decoherence, the role of the environment in such systems is more complex, since it also provides the only source of nonlinearity. We show that even for an unstructured bath,…
We explore the quantum criticality of a two-site model combining quantum Rabi models with hopping interaction. Through a combination of analytical and numerical approaches, we find that the model allows the appearance of a superradiant quantum phase transition (QPT) even in the presence of strong $\mathbf{A}^2$ terms, preventing single-site superradiance. In the two-site model the effect of $\mathbf{A}^2$ terms can be surmounted by the photon delocalization from hopping, and a reversed superradiant QPT occurs as a consequence of the competition between $\mathbf{A}^2$ terms and the hopping inte…
In the framework of the non-secular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions have been obtained beyond the rotating wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong…
We analyze the dynamics of a superconducting qubit and the phenomenon of multiorder Rabi oscillations in the presence of a time-modulated external field. Such a field is also presented as a bichromatic field consisting of two spectral components, which are symmetrically detuned from the qubit resonance frequency. This approach leads to obtaining qualitative quantum effects beyond those for the case of monochromatic excitation of qubits. We calculate Floquet states and quasienergies of the composite system "superconducting qubit plus time-modulated field" for various resonant regimes. We analyz…
We propose an adiabatic protocol for implementing a controlled-phase gate CZ$_θ$ with continuous $θ$ of neutral atoms through a symmetrical two-photon excitation process via the second resonance line, $6P$ in $^{87}$Rb, with a single-temporal-modulation-coupling of the ground state and intermediate state. Relying on different adiabatic paths, the phase factor $θ$ of CZ$_θ$ gate can be accumulated on the logic qubit state $|11\rangle$ alone by calibrating the shape of the temporal pulse where strict zero amplitudes at the start and end of the pulse are not needed. For a wide range of $θ$, we ca…
We present an analytical and comprehensive description of the quantum dynamics of a microwave resonantly driven superconducting phase qubit coupled to a microscopic two-level system (TLS), covering a wide range of the external microwave field strength. Our model predicts several interesting phenomena in such an ac driven four-level bipartite system including anomalous Rabi oscillations, high-contrast beatings of Rabi oscillations, and extraordinary two-photon transitions. Our experimental results in a coupled qubit-TLS system agree quantitatively very well with the predictions of the theoretic…
Recent electron spin resonance experiments on CaWO$_4$:Gd$^{3+}$ and on other magnetic impurities have demonstrated that sustained Rabi oscillations can be created by driving a magnetic moment with a microwave field frequency slightly larger than the Larmor frequency and tuned to the Floquet resonance together with another microwave field (image drive). These observations are confirmed by the new experimental results reported in this paper. We investigate several mechanisms of decoherence and dissipation by using a combination of numerical and analytical techniques. The first microscopic model…
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 propose a set of subradiant states which can be prepared and detected in a one-dimensional optical lattice. We find that the decay rates are highly dependent on the spatial phases imprinted on the atomic chain, which gives systematic investigations of the subradiance in the fluorescence experiments. The time evolution of these states can have long decay time where up to hundred milliseconds of lifetime is predicted for one hundred atoms. They can also show decayed Rabi-like oscillations with a beating frequency determined by the difference of cooperative Lamb shift in the subspace. Experime…
This work introduces the so-called synchrosqueezed wavelet transform, to shed light on the dipolar fluctuations of a thermal ensemble of nuclear spins in a diamond crystal structure, hyperfine-coupled to a central spin. The raw time series of the nuclear spin bath coherent dynamics is acquired through the two-point correlation function computed using the cluster correlation expansion method. The dynamics can be conveniently analyzed according to zero-, single-, and double-quantum transitions derived from the dipolar pairwise spin flips. We show that in the early-time behavior when the coherenc…
We propose an iterative scheme to factor numbers based on the quantum dynamics of an ensemble of interacting bosonic atoms stored in a trap where the single-particle energy spectrum depends logarithmically on the quantum number. When excited by a time-dependent interaction these atoms perform Rabi oscillations between the ground state and an energy state characteristic of the factors. The number to be factored is encoded into the frequency of the sinusoidally modulated interaction. We show that a measurement of the energy of the atoms at a time chosen at random yields the factors with probabil…
We present the model of one-dimensional chain of two-level artificial atoms driven with dc field and quantum light simultaneously in strong coupling regime. The interaction of atoms with light leads to entanglement of electron and photon states (dressing of the atoms). The driving via dc field leads to the Bloch oscillations (BO) in the chain of dressed atoms. We considered the mutual influence of dressing and BO and show that scenario of oscillations dramatically differs from predicted by the Jaynes-Cummings and Bloch-Zener models. We study the evolution of the population inversion, tunneling…
Delayed Formation of Landau Polaritons in Phase-Resolved THz Spectroscopy
Centera Labs, Institute of High Pressure Physics, Polish Academy of Sciences（波兰科学院高压物理研究所）；Centera, Cezamat, Warsaw University of Technology（华沙理工大学）；Weitzman Institute of Science（魏茨曼科学研究所）；Faculty of Physics, University of Warsaw（华沙大学物理学院）
We investigate the dynamics of a driven two-level system (classical Rabi model) using the counter-rotating-hybridized rotating wave method (CHRW), which is a simple method based on a unitary transformation with a parameter $ξ$. This approach is beyond the traditional rotating-wave approximation (Rabi-RWA) and more importantly, remains the RWA form with a renormalized tunneling strength and a modified driving strength. The reformulated rotating wave method not only possesses the same mathematical simplicity as the Rabi-RWA but also allows us to explore the effects of counter-rotating (CR) compo…
The parametric phase-locked oscillator (PPLO), also known as a parametron, is a resonant circuit in which one of the reactances is periodically modulated. It can detect, amplify, and store binary digital signals in the form of two distinct phases of self-oscillation. Indeed, digital computers using PPLOs based on a magnetic ferrite ring or a varactor diode as its fundamental logic element were successfully operated in 1950s and 1960s. More recently, basic bit operations have been demonstrated in an electromechanical resonator, and an Ising machine based on optical PPLOs has been proposed. Here…
The rapidly developing and converging fields of polaritonic chemistry and quantum optics necessitate a unified approach to predict strongly-correlated light-matter interactions with atomic-scale resolution. Combining concepts from both fields presents an opportunity to create a predictive theoretical and computational approach to describe cavity correlated electron-nuclear dynamics from first principles. Towards this overarching goal, we introduce a general time-dependent density-functional theory to study correlated electron, nuclear and photon interactions on the same quantized footing. In o…
