We evaluate the advantages of performing cross-phase modulation (XPM) on a very-far-off-resonance atomic system. We consider a ladder system with a weak (few-photon level) control coherent field imparting a conditional nonlinear phase shift on a probe beam. We find that by coupling to an optical resonator the optimal XPM is enhanced proportional to the finesse of the resonator by a factor of $F/4π$. We present a semi-classical description of the system and show that the phenomenon is optimal in the self-defined condition of off-resonance-effective-cooperativity equal to one.
We consider the Casimir-Polder potential of particles placed inside a metallic cylindrical cavity at finite temperatures, taking account of thermal non-equilibrium effects. In particular, we study how the resonant (thermal non-equilibrium) potential and transition rates can be enhanced by fine-tuning the radius of the cavity to match the transition wavelength of the dominant transition of the particle. Numerical calculations show that the CP potential acting atoms prepared in low-lying Rydberg states can be enhanced beyond 30 kHz, which is within the range of observability of modern experiment…
Broadband homodyne detection of the light transmitted by a Fabry-Perot cavity containing a strongly-coupled $^{133}$Cs atom is used to probe the dynamic optical response in a regime where semiclassical theory predicts bistability but strong quantum corrections should apply. While quantum fluctuations destabilize true equilibrium bistability, our observations confirm the existence of metastable states with finite lifetimes and a hysteretic response is apparent when the optical drive is modulated on comparable timescales. Our experiment elucidates remnant semiclassical behavior in the attojoule …
We present a numerical study comparing semiclassical and quantum models of a damped, strongly interacting cavity QED system composed of a single two-level atom interacting with a single quantized cavity mode driven externally by a tunable monochromatic field. We compute the steady state transmission spectrum of the coupled system under each model and show that in the strong coupling regime, the two models yield starkly different results. The fully quantum mechanical model of the system correctly yields the expected multiphoton transmission spectra while the semiclassical approach results in a …
The transmission of a probe field experiencing electromagnetically induced transparency and optical switching in an atomic medium enclosed in an optical cavity is investigated. Using a semiclassical input-output theory for the interaction between an ensemble of four-level atoms and three optical cavity fields coupled to the same spatial cavity mode, we derive the steady-state transmission spectra of the probe field and discuss the dynamics of the intracavity field buildup. The analytical and numerical results are in good agreement with recent experiments with ion Coulomb crystals [M. Albert et…
We present a simple extension of the semi-classical model for a two-level system in a cavity, in order to incorporate multiple polarized transitions, such as those appearing in neutral and charged quantum dots (QDs), and two nondegenerate linearly polarized cavity modes. We verify the model by exact quantum master equation calculations, and experimentally using a neutral QD in a polarization non-degenerate micro-cavity, in both cases we observe excellent agreement. Finally, the usefulness of this approach is demonstrated by optimizing a single-photon source based on polarization postselection,…
We present detailed numerical simulations of semiclassical and quantum spectra of a cavity quantum electrodynamics system consisting of a single three-level atom in $Λ$-configuration with one of its transitions strongly interacting with a quantized cavity mode while the other is driven by a coherent classical field. After deriving the equations of motion for the expected values of the system operators from the master equation, we compute numerically the semiclassical and quantum spectra of the system under various levels of external driving field strengths. In the semiclassical approach we neg…
We present a systematic semiclassical model for the simulation of the dynamics of a single two-level atom strongly coupled to a driven high-finesse optical cavity. From the Fokker-Planck equation of the combined atom-field Wigner function we derive stochastic differential equations for the atomic motion and the cavity field. The corresponding noise sources exhibit strong correlations between the atomic momentum fluctuations and the noise in the phase quadrature of the cavity field. The model provides an effective tool to investigate localisation effects as well as cooling and trapping times. I…
We present a theoretical study on the nonlinear dynamics and stationary states of an inhomogeneously broadened spin ensemble coupled to a single-mode cavity driven by an external drive with constant amplitude. Assuming a sizeable number of constituents within the ensemble allows us to use a semiclassical approach and to formally reduce the theoretical description to the Maxwell-Bloch equations for the cavity and spin amplitudes. We explore the critical slowing-down effect, quench dynamics, and asymptotic behavior of the system near a steady-state dissipative phase transition accompanied by a b…
We numerically study the dynamics and stationary states of a spin ensemble strongly coupled to a single-mode resonator subjected to loss and external driving. Employing a generalized cumulant expansion approach we analyze finite-size corrections to a semiclassical description of amplitude bistability, which is a paradigm example of a driven-dissipative phase transition. Our theoretical model allows us to include inhomogeneous broadening of the spin ensemble and to capture in which way the quantum corrections approach the semiclassical limit for increasing ensemble size $N$. We set up a criteri…
We benchmark a selection of semiclassical and perturbative dynamics techniques by investigating the correlated evolution of a cavity-bound atomic system to assess their applicability to study problems involving strong light-matter interactions in quantum cavities. The model system of interest features spontaneous emission, interference, and strong coupling behaviour, and necessitates the consideration of vacuum fluctuations and correlated light-matter dynamics. We compare a selection of approximate dynamics approaches including fewest switches surface hopping, multi-trajectory Ehrenfest dynami…
We investigate the dynamics of a two-photon laser under conditions where the spatial variation of the cavity field along the cavity axis is important. Main attention is paid to linear stability analysis and numerical investigation of a two-photon laser for the Maxwell-Bloch equations. The model assumes pumping to the upper state of the two-photon transition. We consider the Maxwell-Bloch equations on the basis of which we study the stability analysis of the steady state of the system. The system is taken to be contained in a ring-laser cavity. Asymptotic expansion of the eigenvalue and analyti…
We experimentally characterize Hopf bifurcation phenomena at femtojoule energy scales in a multi-atom cavity quantum electrodynamical (cavity QED) system, and demonstrate how such behaviors can be exploited in the design of all-optical memory and modulation devices. The data are analyzed using a semiclassical model that explicitly treats heterogeneous coupling of atoms to the cavity mode. Our results highlight the interest of cavity QED systems for ultra-low power photonic signal processing as well as for fundamental studies of mesoscopic nonlinear dynamics.
The dynamics of the Jaynes-Cummings Hamiltonian is studied in the semiclassical limit by reducing it to quantum normal form and deriving the phase space dynamics of classically smooth observables. The resulting systematic semiclassical approximation is used to analyze the Rabi oscillations of a photon wave packet revealing subtle consequences of the uncertainty principle.
We study the general-setting quantum geometric phase acquired by a particle in a vibrating cavity. Solving the two-level theory with the rotating-wave approximation and the SU(2) method, we obtain analytic formulae that give excellent descriptions of the geometric phase, energy, and wavefunction of the resonating system. In particular, we observe a sudden $π$-jump in the geometric phase when the system is in resonance. We found similar behaviors in the geometric phase of a spin-1/2 particle in a rotating magnetic field, for which we developed a geometrical model to help visualize its evolution…
The spectral and temporal response of a set of $N$ quantum emitters embedded in a photonic cavity is studied. Quantum mechanically, such systems can be described by the Tavis-Cummings (TC) model of $N$ two-level systems coupled to a single light mode. Here we compare the full quantum solution of the TC model for different numbers of quantum emitters with its semiclassical limit after a pulsed excitation of the cavity mode. Considering different pulse amplitudes, we find that the spectra obtained from the TC model approach the semiclassical one for an increasing number of emitters $N$. Furtherm…
We complement previous studies of an ion coupled with an optical cavity in the dispersive regime, for a model which exhibits bistability of different configurations in the semiclassical description. Our approach is based on a truncated evolution in phase space and is intended to explore an especially interesting parameter region where the fully quantum-mechanical solution becomes hard but the crudest semiclassical approach fails to capture essential phenomena. We compare the results of our techniques with the ones from numerical diagonalization of the quantum evolution and find that although t…
We analyse the dynamics leading to radiative cooling of an atomic ensemble confined inside an optical cavity when the atomic dipolar transitions are incoherently pumped and can synchronize. Our study is performed in the semiclassical regime and assumes that cavity decay is the largest rate in the system dynamics. We identify three regimes characterising the cooling. At first hot atoms are individually cooled by the cavity friction forces. After this stage, the atoms' center-of-mass motion is further cooled by the coupling to the internal degrees of freedom while the dipoles synchronize. In the…
We address frequency-dependent quantum transport through mesoscopic conductors in the semiclassical limit. By generalizing the trajectory-based semiclassical theory of dc quantum transport to the ac case, we derive the average screened conductance as well as ac weak-localization corrections for chaotic conductors. Thereby we confirm respective random matrix results and generalize them by accounting for Ehrenfest time effects. We consider the case of a cavity connected through many leads to a macroscopic circuit which contains ac-sources. In addition to the reservoir the cavity itself is capaci…
We consider an optomechanical system composed by a trapped ion dispersively coupled to a single mode of a pumped optical cavity. We focus in a parameter range for which the semiclassical description predicts two clearly distinct equilibrium configurations in the limits of small and large photon pumping, while a bistable regime is found for intermediate pumping. This semiclassical description, however, is not valid in close proximity of the system transitions or when the mean photon number is low. Here we provide a numerical analysis of the fully quantum state in the few-photon regime, explorin…
We address a fundamental issue in quantum mechanics and quantum information theory, the generation of an entangled pair of qubits that interact solely through a third, semiclassical degree of freedom, in the framework of cavity quantum electrodynamics. We show that finite, though not maximal, entanglement is obtainable in the classical limit, at the price of a diverging effective interaction time. The optimal atomic entanglement derives from a trade-off between the atomic entanglement in a sub-wave packet and the purity of the atomic state. Decoherence by photon loss sets an upper limit on the…
We extend an earlier semiclassical model to describe the dissipative motion of N atoms coupled to M modes inside a coherently driven high-finesse cavity. The description includes momentum diffusion via spontaneous emission and cavity decay. Simple analytical formulas for the steady-state temperature and the cooling time for a single atom are derived and show surprisingly good agreement with direct stochastic simulations of the semiclassical equations for N atoms with properly scaled parameters. A thorough comparison with standard free-space Doppler cooling is performed and yields a lower tempe…
The semiclassical dynamics of atoms are theoretically studied, when the atoms are confined inside a standing-wave high-finesse resonator. The atoms are cooled by scattering processes in which the photons of a transverse laser are coherently scattered into the cavity mode. We derive a Fokker-Planck equation for the atomic center-of-mass variables which allows us to determine the equations of motion in the semiclassical limit for any value of the intensity of the laser field. We extract its prediction for the dynamics when the resonator is essentially in the vacuum state and the atoms are cooled…
Giant optical Faraday rotation (GFR) and giant optical circular birefringence (GCB) induced by a single quantum-dot spin in an optical microcavity can be regarded as linear effects in the weak-excitation approximation if the input field lies in the low-power limit [Hu et al, Phys.Rev. B {\bf 78}, 085307(2008) and ibid {\bf 80}, 205326(2009)]. In this work, we investigate the transition from the weak-excitation approximation moving into the saturation regime comparing a semiclassical approximation with the numerical results from a quantum optics toolbox [S.M. Tan, J. Opt. B {\bf 1}, 424 (1999)]…
We study the effect of spontaneous emission and incoherent atomic pumping on the nonlinear semiclassical dynamics of the unbalanced Dicke model -- a generalization of the Dicke model that features independent coupling strengths for the co- and counter-rotating interaction terms. As well as the ubiquitous superradiant behavior the Dicke model is well-known for, the addition of spontaneous emission combined with the presence of strong counter-rotating terms creates laser-like behavior termed counter-lasing. These states appear in the semiclassical model as stable periodic orbits. We perform a co…
We theoretically investigated the ground states of coupled arrays of cavity quantum electrodynamical (cavity QED) systems in presence of two photon modes. Within the Gutzwiller-type variational approach, we found the first-order quantum phase transition between Mott insulating and superfluid phases as well as the conventional second-order one. The first-order phase transition was found only for specific types of emitter models, and its physical origin is clarified based on the analytic arguments which are allowed in the perturbative and semiclassical limits. The first-order transition of the c…
A nonlinear stimulated Raman adiabatic passage (STIRAP) is a fascinating physical process that dynamically explores chaotic and non-chaotic phases. In a recent paper Phys. Rev. Res. 2, 042004 (R) (2020), such a phenomenon is realized in a cavity-QED platform. There, the emergence of chaos and its impact on STIRAP efficiency are mainly demonstrated in the semiclassical limit. In the present paper I treat the problem in a fully quantum many-body framework. With the aim of extracting quantum signatures of a classically chaotic system, it is shown that an out-of-time-ordered correlator (OTOC) meas…
A semiclassical model is presented for characterizing the linear response of elementary quantum optical systems involving cavities, optical fibers, and atoms. Formulating the transmission and reflection spectra using a scattering-wave (transfer matrix) approach, the calculations become easily scalable. To demonstrate how useful this method is, we consider the example of a simple quantum network, i.e., two cavity-QED systems connected via an optical fiber. Differences between our quasi-exact transfer matrix approach and a single-mode, linearized quantum-optical model are demonstrated for parame…
We propose an experimental realization of a time crystal using an atomic Bose-Einstein condensate in a high finesse optical cavity pumped with laser light detuned to the blue side of the relevant atomic resonance. By mapping out the dynamical phase diagram, we identify regions in parameter space showing stable limit cycle dynamics. Since the model describing the system is time-independent, the emergence of a limit cycle phase indicates the breaking of continuous time translation symmetry. Employing a semiclassical analysis to demonstrate the robustness of the limit cycles against perturbations…
Observation of quantum phenomena in cryogenic, optically cooled mechanical resonators has been recently achieved by a few experiments based on cavity optomechanics. A well-established experimental platform is based on a thin film stoichiometric ($ Si_3 N_4 $) nanomembrane embedded in a Fabry-Perot cavity, where the coupling with the light field is provided by the radiation pressure of the light impinging on the membrane surface. Two crucial parameters have to be optimized to ensure that these systems work at the quantum level: the cooperativity $ C$ describing the optomechanical coupling and t…
Discrete time crystals are a recently proposed and experimentally observed out-of-equilibrium dynamical phase of Floquet systems, where the stroboscopic evolution of a local observable repeats itself at an integer multiple of the driving period. We address this issue in a driven-dissipative setup, focusing on the modulated open Dicke model, which can be implemented by cavity or circuit QED systems. In the thermodynamic limit, we employ semiclassical approaches and find rich dynamical phases on top of the discrete time-crystalline order. In a deep quantum regime with few qubits, we find clear s…
The setting of Moessbauer nuclei embedded in thin-film cavities has facilitated an aspiring platform for x-ray quantum optics as shown in several recent experiments. Here, we generalize the theoretical model of this platform that we developed earlier [Phys. Rev. A 88, 043828 (2013)]. The theory description is extended to cover multiple nuclear ensembles and multiple modes in the cavity. While the extensions separately do not lead to qualitatively new features, their combination gives rise to cooperative effects between the different nuclear ensembles and distinct spectral signatures in the obs…
We study a system consisting of a superconducting flux qubit strongly coupled to a microwave cavity. Externally applied qubit driving is employed in order to manipulate the spectrum of dressed states. We observe resonance narrowing in the region where the splitting between the two dressed fundamental resonances is tuned to zero. The narrowing in this region of overlapping resonances can be exploited for long-time storage of quantum states. In addition, we measure the response to strong cavity mode driving, and find a qualitative deviation between the experimental results and the predictions of…
We theoretically analyze the collective dynamics of a thermal beam of atomic dipoles that couple to a single mode when traversing an optical cavity. For this setup we derive a semiclassical model and determine the onset of superradiant emission and its stability. We derive analytical expressions for the linewidth of the emitted light and compare them with numerical simulations. In addition, we find and predict two different superradiant phases; a steady-state superradiant phase and a multi-component superradiant phase. In the latter case we observe sidebands in the frequency spectrum that can …
We demonstrate the emergence of a time crystal of atoms in a high-finesse optical cavity driven by a phase-modulated transverse pump field, resulting in a shaken lattice. This shaken system exhibits macroscopic oscillations in the number of cavity photons and order parameters at noninteger multiples of the driving period, which signals the appearance of an incommensurate time crystal. The subharmonic oscillatory motion corresponds to dynamical switching between symmetry-broken states, which are nonequilibrium bond ordered density wave states. Employing a semiclassical phase-space representatio…
A cold atomic gas with an inverted population on a transition coupled to a field mode of an optical resonator constitutes a generic model of a laser. For quasi-continuous operation, external pumping, trapping and cooling of the atoms is required to confine them in order to achieve enough gain inside the resonator. As inverted atoms are high-field seekers in blue detuned light fields, tuning the cavity mode to the blue side of the atomic gain transition allows for combining lasing with stimulated cavity cooling and dipole trapping of the atoms at the antinodes of the laser field. We study such …
At strong pump powers, a semiconductor optical cavity passes through a Hopf bifurcation and undergoes self-oscillation. We simulate this device using semiclassical Langevin equations and assess the effect of quantum fluctuations on the dynamics. Below threshold, the cavity acts as a phase-insensitive linear amplifier, with noise $\sim 5\times$ larger than the Caves bound. Above threshold, the limit cycle acts as an analog memory, and the phase diffusion is $\sim 10\times$ larger than the bound set by the standard quantum limit. We also simulate entrainment of this oscillator and propose an opt…
Quantum phase transitions of the Cavity Heisenberg spin-chain
College of Physics and Electronic Engineering, Northwest Normal University（西北师范大学物理与电子工程学院）；Gansu Provincial Research Center for Basic Disciplines of Quantum Physics（甘肃省量子物理基础学科研究中心）
Pattern formation of atoms in high-finesse optical resonators results from the mechanical forces of light associated with superradiant scattering into the cavity mode. It occurs when the laser intensity exceeds a threshold value, such that the pumping processes counteract the losses. We consider atoms driven by a laser and coupling with a mode of a standing-wave cavity and describe their dynamics with a Fokker-Planck equation, in which the atomic motion is semiclassical but the cavity field is a full quantum variable. The asymptotic state of the atoms is a thermal state, whose temperature is s…
The interaction of a quantized electromagnetic field in a cavity with a set of two-level atoms inside can be described with algebraic Hamiltonians of increasing complexity, from the Rabi to the Dicke models. Their algebraic character allows, through the use of coherente states, a semiclassical description in phase space, where the non-integrable Dicke model has regions associated with regular and chaotic motion. The appearance of classical chaos can be quantified calculating the largest Lyapunov exponent in the whole available phase space for a given energy. In the quantum regime, employing ef…
We study Winter or delta-shell model at finite volume (length), describing a small resonating cavity weakly-coupled to a large one. For generic values of the coupling, a resonance of the usual model corresponds, in the finite-volume case, to a compression of the spectral lines; for specific values of the coupling, a resonance corresponds instead to a degenerate or a quasi-degenerate doublet. A secular term of the form g^3 N occurs in the perturbative expansion of the momenta (or of the energies) of the particle at third order in g, where g is the coupling among the cavities and N is the ratio …
We highlight the importance of quantum fluctuations in organizing a dissipative quantum phase transition for the driven Jaynes-Cummings interaction with variable qubit-cavity detuning. The system response presents a substantial difference from the predictions of the semiclassical theory, the extent of which is revealed in the properties of quantum bistability, and visualized with the help of quasi-distribution functions for the cavity field, subject to an appropriate scale parameter. States anticipated by the neoclassical theory of radiation coexist in the quantum picture, following the occurr…
