We have investigated vacuum Rabi oscillation of an atom coupled with single-mode cavity field exactly, and compared the results with that of J-C model. The results show that, there is damping Rabi oscillation for an atom, even in strong coupling regime. For small detuning and weak coupling case, the probability for the atom in excited state oscillates against time with different frequency and amplitude from that of J-C model. It exhibits the counter-rotating wave interaction could significantly effect the dynamic behavior of the atom, even under the condition in which the RWA is considered to be justified. On the other hand, the results also reveal that the counter-rotating wave interaction could obviously modify the decay rate in strong coupling regime. And there is Rabi oscillation for initially unexcited atom, which is contrary to that of J-C model.
In this article we apply algebraic operator ordering property of the basic atom and field operators to construct and prove conservation of the excitation number operator of the anti-rotating (anti-Jaynes-Cummings) component of the quantum Rabi model. The corresponding U(1) and parity symmetry operators have also been derived in explicit form.
We present an interaction scheme to control counter-rotating terms in the quantum Rabi model. We show that by applying a sequence of $π/2$ phase kicks to a two-level atom and a single mode quantized field, the natural dynamics of the Rabi model can be interrupted in a way that counter-rotating transitions can be significantly enhanced. This is achieved by a suitable timing of the phase kicks determined by a phase matching condition. If the time between successive kicks is sufficiently short, our scheme is turned into a dynamical decoupling problem in which the effects of counter-rotating terms…
We present a novel approach called the intermediate rotating wave approximation (IRWA), which employs a time-averaging method to encapsulate the dynamics of light-matter interaction from strong to ultrastrong coupling regime. In contrast to the ordinary rotating wave approximation, this method addresses the co-rotating and counter-rotating terms separately to trace their physical consequences individually, and thus establishes the continuity between the Jaynes-Cummings model and the quantum Rabi model. We investigate IRWA in near resonance and large detuning cases. Our IRWA not only agrees wel…
The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In …
he effect of the counter-rotating terms in the Jaynes-Cummings model is investigated with an extended coherent-state approach. The counter-rotating terms greatly modify the vacuum Rabi splitting. Two peaks with different heights in the weak coupling regime and more than two peaks in the intermediate coupling regime are predicted. The collapses and revivals in the evolution of the atomic population inversion disappear in the intermediate coupling regime, but reappear in the strong coupling regime. This reappearance is similar to that under the rotating-wave approximation, attributed to the summ…
This article is a response to the continued assumption, cited even in reports and reviews of recent experimental breakthroughs and advances in theoretical methods, that the antiJaynes-Cummings (AJC) interaction is an intractable energy non-conserving component of the quantum Rabi model (QRM). We present three key features of QRM dynamics : (a) the AJC interaction component has a conserved excitation number operator and is exactly solvable (b) QRM dynamical space consists of a rotating frame (RF) dominated by an exactly solved Jaynes-Cummings (JC) interaction specified by a conserved JC excitat…
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.
We study quantum optics with the atoms coupled to the quantized electromagnetic (EM) field in a non-inertial reference frame by making use of quantum field theory in curved spacetime. We rigorously establish the microscopic model for a two-level atom interacting with the quantized EM field in a rotating ring cavity by deriving a Jaynes-Cummings (JC) type Hamiltonian. Due to the two fold degeneracy of the ring cavity modes, the Rabi splitting exhibits three rather than two resonant frequency peaks. We find that the heights of the two side peaks show a sensitive linear dependence on the rotating…
Generalized squeezing rotating-wave approximation (GSRWA) is proposed by employing both the displacement and the squeezing transformations. A solvable Hamiltonian is reformulated in the same form as the ordinary RWA ones. For a qubit coupled to oscillators experiment, a well-defined Schrödinger-cat-like entangled state is given by the displaced-squeezed oscillator state instead of the original displaced state. For the isotropic Rabi case, the mean photon number and the ground-state energy are expressed analytically with additional squeezing terms, exhibiting a substantial improvement of the GS…
We present an interaction scheme that exhibits a dynamical consequence of virtual photons carried by a vacuum-field dressed two-level atom in the ultrastrong coupling regime. We show that, with the aid of an external driving field, virtual photons provide a transition matrix element that enables the atom to evolve coherently and reversibly to an auxiliary level accompanied by the emission of a real photon. The process corresponds to a type of vacuum Rabi oscillation, and we show that the effective vacuum Rabi frequency is proportional to the amplitude of a single virtual photon in the ground s…
Comparison of theory of Rabi oscillations with experiment [M. Wilczewski and M. Czachor, Phys. Rev. A {\bf 79}, 033836 (2009)] suggests that cavity lifetime parameters obtained in measurements with many photons may be much smaller than those applicable to almost vacuum states of light. In this context we show that the conclusion remains unchanged even if one takes a more realistic description of the initial state of light in cavity.
We predict large cooperative effect involving two atom two photon vacuum Rabi oscillations in a high quality cavity. The two photon emission occurs as a result of simultaneous de-excitation of both atoms with two photon resonance condition $ω_1+ω_2\approx ω_a+ω_b$, where $ω_1$, $ω_2$ are the atomic transition frequencies and $ω_a$, $ω_b$ are the frequencies of the emitted photons. The actual resonance condition depends on the vacuum Rabi couplings. The effect can be realized either with identical atoms in a bimodal cavity or with nonidentical atoms in a single mode cavity.
We investigate the spectrum of energy and eigenstates of a hybrid cavity optomechanical system, where a cavity field mode interacts with a mechanical mode of a vibrating end mirror via radiation pressure and with a two level atom via electric dipole interaction. In the spirit of approximations developed for the quantum Rabi model beyond rotating-wave approximation (RWA), we implement the so-called generalized RWA (GRWA) to diagonalize the tripartite Hamiltonian for arbitrary large couplings. Notably, the GRWA approach still allows to rewrite the hybrid Hamiltonian in a bipartite form, like a R…
Here we use perturbation techniques based on the averaging method to investigate Rabi oscillations in cw and pulse-driven two-level systems (TLS's). By going beyond the rotating-wave approximation, especifically to second-order in perturbation, we obtain the Bloch-Siegert shift of the TLS resonant frequency, in which the resonant frequency increases with the driving field amplitude. This frequency shift implies that short resonant $π$-pulses in which the Rabi frequency is approximately 40\% or higher of the transition frequency do not achieve complete inversion in TLS's. Hence, guided by analy…
The Rabi Hamiltonian describes a single mode of electromagnetic radiation interacting with a two-level atom. Using the coupled cluster method, we investigate the time evolution of this system from an initially empty field mode and an unexcited atom. We give results for the atomic inversion and field occupation, and find that the virtual processes cause the field to be squeezed. No anti-bunching occurs.
We propose a practical approach to manipulate the counter-rotating (CR) interactions in the quantum Rabi model by introducing a sinusoidal modulation to the transition frequency of the quantum two-level system in this model. By choosing appropriate modulation frequency and amplitude, enhancement and suppression of the CR interactions can be achieved in the Jaynes-Cummings regime (including both weak- and strong-coupling cases) as well as the ultrastrong-coupling regime. In particular, we calculate the output photon emission of the cavity vacuum state under enhanced CR terms. Our results show t…
Considering the intrinsic nonlinearity in a nanomechanical resonator coupled to a charge qubit, vacuum Rabi splitting effect is studied in a nanomechanical QED (qubit-resonator) system. A driven nonlinear Jaynes-Cummings model describes the dynamics of this qubit-resonator system. Using quantum regression theorem and master equation approach, we have calculated the two-time correlation spectrum analytically. In the weak driving limit, these analytical results clarify the influence of the driving strength and nonlinearity parameter on the correlation spectrum. Also, numerical calculations confi…
The transmission spectrum for one atom strongly coupled to the field of a high-finesse optical resonator is observed to exhibit a clearly resolved vacuum-Rabi splitting characteristic of the normal modes in the eigenvalue spectrum of the atom-cavity system. A new Raman scheme for cooling atomic motion along the cavity axis enables a complete spectrum to be recorded for an individual atom trapped within the cavity mode, in contrast to all previous measurements in cavity QED that have required averaging over many atoms.
Squeezed states of harmonic oscillators are a central resource for continuous-variable quantum sensing, computation and communication. Here we propose a method for the generation of very good approximations to highly squeezed vacuum states with low excess anti-squeezing using only a few oscillator-qubit coupling gates through a Rabi-type interaction Hamiltonian. This interaction can be implemented with several different methods, which has previously been demonstrated in superconducting circuit and trapped-ion platforms. The protocol is compatible with other protocols manipulating quantum harmo…
We present an analytical method for the two-qubit quantum Rabi model. While still operating in the frame of the generalized rotating-wave approximation (GRWA), our method further embraces the idea of introducing variational parameters. The optimal value of the variational parameter is determined by minimizing the energy function of the ground state. Comparing with numerical exact results, we show that our method evidently improves the accuracy of the conventional GRWA in calculating fundamental physical quantities, such as energy spectra, mean photon number, and dynamics. Interestingly, the ac…
The paper continues the analysis of vacuum Rabi oscillations we started in Part I [Phys. Rev. A {\bf 79}, 033836 (2009)]. Here we concentrate on experimental consequences for cavity QED of two different classes of representations of harmonic oscillator Lie algebras. The zero-temperature master equation, derived in Part I for irreducible representations of the algebra, is reformulated in a reducible representation that models electromagnetic fields by a gas of harmonic oscillator wave packets. The representation is known to introduce automatic regularizations that in irreducible representations…
Environmental influences on the dynamics of a coupled qubit-oscillator system are studied analytically. We investigate the case of a quasi-degenerate qubit within the ultra-strong coupling regime for which the qubit frequency is much smaller than the frequency of the oscillator, and the coupling between the qubit and the oscillator is large, both of which invalidate the usually employed rotating wave approximation. In contrast to the standard quantum optics master equation, we explicitly take the qubit-oscillator coupling into account while microscopically deriving a dressed state master equat…
In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called $G$-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order a…
For two discrete-level quantum systems in interaction, we follow the displacement in the complex plane of the eigen-energies of the compound system when the spectrum of one of the two systems becomes continuous. These new points are usually called resonances. This allows us to define and to calculate a critical value of the coupling constant which separates two well-known coupling regimes. We also give an example of these resonances for the hydrogen atom coupled to the continuum of the states of the transverse electromagnetic field in the vacuum. We justify that some resonances be neglected.
The interaction between an atom and a one mode external driving field is an ubiquitous problem in many branches of physics and is often modeled using the Rabi Hamiltonian. In this paper we present a series of analytically solvable Hamiltonians that approximate the Rabi Hamiltonian and compare our results to the Jaynes-Cummings model which neglects the so-called counter-rotating term in the Rabi Hamiltonian. Through a unitary transformation that diagonlizes the Jaynes-Cummings model, we transform the counter-rotating term into separate terms representing several different physical processes. By…
We study the all-optical time-control of the strong coupling between a single cascade three-level quantum emitter and a microcavity. We find that only specific arrival-times of the control pulses succeed in switching-off the Rabi oscillations. Depending on the arrival times of control pulses, a variety of exotic non-adiabatic cavity quantum electrodynamics effects can be observed. We show that only control pulses with specific arrival times are able to suddenly switch-off and -on first-order coherence of cavity photons, without affecting their strong coupling population dynamics. Such behavior…
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…
In this paper, we propose a theoretical scheme to detect the emitter-resonator coupling strength in the ultra-strong coupling regime in the quantum Rabi model via introducing an auxiliary resonator. We demonstrate the total system as a two-mode Rabi model and obtain the ground state by the transformed rotating wave approximation, which is shown to be superior to the usually applied rotating wave approximation. Here, the coupling strength is detected by monitoring the average excitation number in the auxiliary resonator and the sensitivity of the detection scheme is discussed analytically.
We define the anisotropic Rabi model as the generalization of the spin-boson Rabi model: The Hamiltonian system breaks the parity symmetry; the rotating and counter-rotating interactions are governed by two different coupling constants; a further parameter introduces a phase factor in the counter-rotating terms. The exact energy spectrum and eigenstates of the generalized model is worked out. The solution is obtained as an elaboration of a recent proposed method for the isotropic limit of the model. In this way, we provide a long sought solution of a cascade of models with immediate relevance …
We study numerically the dynamics of the Rabi Hamiltonian, describing the interaction of a single cavity mode and a two-level atom without the rotating wave approximation, subjected to damping and dephasing reservoirs included via usual Lindblad superoperators in the master equation. We show that the combination of the antirotating term and the atomic dephasing leads to linear asymptotic photons generation from vacuum. We reveal the origins of the phenomenon and estimate its importance in realistic situations.
Spin squeezing by geometric focusing in vacuum Rabi oscillations
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料中心）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学量子器件北京市重点实验室）；National Laboratory of Solid State Microstructures and School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学固体微结构物理国家重点实验室、物理学院、现代工程与应用科学协同创新中心）；MIT-Harvard Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology（麻省理工学院电子研究实验室与MIT-哈佛超冷原子中心）
Considering well localized atom, the mean field theory (MFT) was applied to replace the operators by equivalent expectation values. The Rabi model was reduced to a fourth orders NDE describing atoms position. Solution by the harmonic balance method (HBM) showed good accuracy and consistency to the numerical results, which introduces it as a useful tool in the quantum dynamics studies.
We introduce the anisotropic two-photon Rabi model in which the rotating and counter rotating terms enters along with two different coupling constants. Eigenvalues and eigenvectors are studied with exact means. We employ a variation of the Braak method based on Bogolubov rotation of the underlying $su(1,1)$ Lie algebra. Accordingly, the spectrum is provided by the analytical properties of a suitable meromorphic function. Our formalism applies to the two-modes Rabi model as well, sharing the same algebraic structure of the two-photon model. Through the analysis of the spectrum, we discover that…
We report the observation of the vacuum Rabi splitting of a single quantum emitter by measuring its direct spontaneous emission into free space. We used a semiconductor quantum dot inside a photonic crystal nanocavity, in conjunction with an appropriate cavity design and filtering with a polarizer and an aperture, enabling the extraction of the inherently-weak emitter's signal. The emitter's vacuum Rabi spectra exhibit clear differences to those measured by detecting the cavity photon leakage. Moreover, we observed an asymmetric vacuum Rabi spectrum induced by interference between the emitter …
The Rabi Hamiltonian, describing the interaction between a two-level atomic system and a single cavity mode of the electromagnetic field, is one of the fundamental models in quantum optics. The model becomes exactly solvable by considering an atom without permanent dipole moments, whose excitation energy is quasi-resonant with the cavity photon energy, and by neglecting the non resonant (counter-rotating) terms. In this case, after including the decay of either the atom or the cavity mode to a continuum, one is able to derive the well-known phenomenology of quasi-resonant transitions, includin…
We report time-domain observation of vacuum Rabi oscillations in a single quantum dot strongly coupled to a nanocavity under incoherent optical carrier injection. We realize a photonic crystal nanocavity with a very high quality factor of >80,000 and employ it to clearly resolve the ultrafast vacuum Rabi oscillations by simple photoluminescence-based experiments. We found that the time-domain vacuum Rabi oscillations were largely modified when changing the pump wavelength and intensity, even when marginal changes were detected in the corresponding photoluminescence spectra. We analyze the meas…
We demonstrate the emergence of selective $k$-photon interactions in the strong and ultrastrong coupling regimes of the quantum Rabi model with a Stark coupling term. In particular, we show that the interplay between the rotating and counter-rotating terms produces multi-photon interactions whose resonance frequencies depend, due to the Stark term, on the state of the bosonic mode. We develop an analytical framework to explain these $k$-photon interactions by using time-dependent perturbation theory. Finally, we propose a method to achieve the quantum simulation of the quantum Rabi model with …
The demonstration of enhanced spontaneous emission of nanoscaled optical emitters near metallic nanoparticles and the recent realization of a nanolaser based on surface plasmon amplification by stimulated emission of radiation (spaser) encourage the search for strong coupling regime at the nanoscale. Here we propose the concept of nanopolaritons. We demonstrate with accurate scattering calculations that the strong coupling regime of a single quantum emitter (a semiconductor quantum dot) placed in the gap between two metallic nanoparticles can be achieved. The largest dimension of the investiga…
Toida et al. claim in their recent article [Phys. Rev. Lett. 110, 066802 (2013)] that they `report a direct observation of vacuum Rabi splitting in a GaAs/AlGaAs double quantum dot (DQD) based charge qubit coupled with a superconducting coplanar waveguide (CPW) resonator'. In this comment, we challenge the main claims made in their paper and show that their results: a) do not provide any evidence of vacuum Rabi oscillations and b) do not provide any direct evidence of vacuum Rabi splitting.
In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counter-rotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the original model. We derive the corresponding $G$-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities and are doubly degenerate. We find analytically that the ground-state and the first excited state can cross several times, indicat…
When an atom strongly couples to a cavity, it can undergo coherent vacuum Rabi oscillations. Controlling these oscillatory dynamics quickly relative to the vacuum Rabi frequency enables remarkable capabilities such as Fock state generation and deterministic synthesis of quantum states of light, as demonstrated using microwave frequency devices. At optical frequencies, however, dynamical control of single-atom vacuum Rabi oscillations remains challenging. Here, we demonstrate coherent transfer of optical frequency excitation between a single quantum dot and a cavity by controlling vacuum Rabi o…
I revisit a longstanding question in quantum optics; When is the rotating wave approximation justified? In terms of the Jaynes-Cummings and Rabi models I demonstrate that the approximation in general breaks down in the adiabatic limit regardless of system parameters. This is explicitly shown by comparing Berry phases of the two models, where it is found that this geometrical phase is strictly zero in the Rabi model contrary to the non-trivial Berry phase of the Jaynes-Cummings model. The source of this surprising result is traced back to different topologies in the two models.
A generalized quantum Rabi Hamiltonian with both one- and two-photon terms has emerged in the circuit quantum electrodynamics system for a decade. The usual parity symmetry is broken naturally in the simultaneous presence of both couplings, which complicates analytical treatments, even in the rotating wave approximations. In this paper, we propose an adiabatic approximation to this generic model by using Bogoliubov operators, and obtain a very concise analytical solution for both eigenvalues and eigenstates. Although the adiabatic approximation is only exact in the vanishing limit of the qubit…
We show that it is possible to realize quantum superpositions of switched-on and -off strong light-matter interaction in a single quantum dot- semiconductor microcavity system. Such superpositions enable the observation of counterintuitive quantum conditional dynamics effects. Situations are possible where cavity photons as well as the emitter luminescence display exponential decay but their joint detection probability exhibits vacuum Rabi oscillations. Remarkably, these quantum correlations are also present in the nonequilibrium steady state spectra of such coherently driven dissipative quant…
Two-photon sideband transition in a driven quantum Rabi model : Quantitative discussions with derived longitudinal drives and beyond the rotating wave approximation
The 1996 Brune {\it et al.} experiment on vacuum Rabi oscillation is analyzed by means of alternative models of atom-reservoir interaction. Agreement with experimental Rabi oscillation data can be obtained if one defines jump operators in the dressed-state basis, and takes into account thermal fluctuations between dressed states belonging to the same manifold. Such low-frequency transitions could be ignored in a closed cavity, but the cavity employed in the experiment was open, which justifies our assumption. The cavity quality factor corresponding to the data is $Q=3.31\cdot 10^{10}$, whereas…
Various quantum phase transitions in the anisotropic Rabi-Stark model with both the nonlinear Stark coupling and the linear dipole coupling between a two-level system and a single-mode cavity are studied in this work. The first-order quantum phase transitions are detected by the level crossing of the ground-state and the first-excited state with the help of the pole structure of the transcendental functions derived by the Bogoliubov operators approach. As the nonlinear Stark coupling is the same as the cavity frequency, this model can be solved by mapping to an effective quantum oscillator. Al…
We study two models describing the interaction of a two-level system with two quantum field modes. The first one is equivalent to a dissipative two-state system with just two boson fields in the absence of tunneling. The second describes two orthogonal fields interacting with the corresponding orthogonal dipoles of a two-level system. We show that both models present a partial two-mode $SU(2)$ symmetry and that they can be solved in the exceptional case of resonant fields. We study their ground state configurations, that is, we find the quantum precursors of the corresponding semi-classical ph…
We present OpenMP version of a Fortran program for solving the Gross-Pitaevskii equation for a harmonically trapped three-component rotating spin-1 spinor Bose-Einstein condensate (BEC) in two spatial dimensions with or without spin-orbit (SO) and Rabi couplings. The program uses either Rashba or Dresselhaus SO coupling. We use the split-step Crank-Nicolson discretization scheme for imaginary- and real-time propagation to calculate stationary states and BEC dynamics, respectively.
Controllable interaction between photons and distant spins via vacuum Rabi oscillations
QuTech and Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学量子纳米科学卡弗里研究所）；Netherlands Organization for Applied Scientific Research (TNO)（荷兰应用科学研究组织）
When an atom is strongly coupled to a cavity, the two systems can exchange a single photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED has been reached experimentally where the strength of the interaction between light and artificial atoms (qubits) becomes comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modif…
We investigate first- and second-order quantum phase transitions of the anisotropic quantum Rabi model, in which the rotating- and counter-rotating terms are allowed to have different coupling strength. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches we extract the phase diagram, scaling functions, and critical exponents, which allows us to establish that the universality class at finite? anisotropy is the same as the isotropic limit. We also reveal other interesting features, including a superradian…
We study the coherent effect of the Casimir-Polder interaction on the oscillations of two-photon driven atoms. We find that, for oscillations between two degenerate states in lambda-configuration, shifts on the Rabi frequency may be induced by non-additive level shifts. For oscillations between two Rydberg states in ladder-configuration, shifts on the Rabi frequency may be induced by the effective renormalization of the laser interaction.
We investigate the applicability of the two major approximations which are most commonly employed in the study of the quantum Rabi model, namely the description of a resonant cavity mode as a single-mode quantized field and the use of the rotating wave approximation. Starting from the Hamiltonian of a two-level system interacting with a multi-mode quantized field, we perform the canonical transformation of the field operators. This allows one to partition the Hamiltonian of the system into two parts. The first part is the interaction of the two-level system with a single collective field mode,…
