Spectral collapse, the transition from discrete to continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model in optical phase space and find that, in the so-called degenerate qubit regime, the collapse is similar to that happening in the transition from a quantum harmonic to an inverted quadratic potential with the free-partical potential as transition point. In this regime, it is possible to construct Dirac-normalizable eigenfunctions for the model that show well defined parity. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs.
This paper presents a proof of the existence of novel bound states of the two-photon quantum Rabi model at the collapse point. The two-photon Rabi model is interesting not only for its important role on non-linear light-matter interaction, but also for the exhibition of many-energy-levels degenerating process called the "spectral collapse". The squeezing property of the two-photon annihilation and creation operators is the origin for this phenomenon which is well studied without the energy-slitting term $ω_0$. However, many numerical studies have pointed out that with the presence of $ω_0$ , s…
The analytically exact solutions to the two-photon Rabi-Stark model are given by using the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived. The zeros of the transcendental functions reproduce completely the regular spectra. The first-order quantum phase transitions are also detected by the pole structure of the derived transcendental functions, similar to the one-photon Rabi-Stark model, different from the two-photon Rabi model where the energy levels for the ground-state and the first excited state do not cross. The spectra collapse char…
We show analytically that the collapse and revival in the population dynamics of the atom-cavity coupled system under the rotating wave approximation (RWA), valid only at very weak coupling, is an artifact as the atom-cavity coupling is increased. Even the first-order correction to the RWA is able to bring about the absence of the collapse in the dynamics of atomic population inversion thanks to intrinsic oscillations resulting from the transitions between two levels with the same atomic quantum number. The removal of the collapse is valid for a wide range of coupling strengths which are acces…
A double rephasing photon echo is analyzed for inversion-free photon echo-based quantum memories using controlled Rabi flopping, where the Rabi flopping is used for phase control of atom coherence. Unlike the rephasing-caused pi-phase shift in a single rephasing scheme, the control Rabi flopping between the excited state and an auxiliary third state induces coherence inversion. Thus, the absorptive photon echo in a double rephasing scheme can be manipulated to be emissive via the controlled Rabi flopping. Here we present a quantum coherence control of atom phases in a double rephasing photon e…
The two-photon quantum Rabi model with quadratic coupling is studied using extended squeezed states and we derive $G$-functions for Bargmann index $q=1/4$ and $3/4$. The simple singularity structure of the $G$-function allows to draw conclusions about the distribution of eigenvalues along the real axis. The previously found picture of the spectral collapse at critical coupling $g_{\mathrm{c}}$ has to be modified regarding the low lying states, especially the ground state: We obtain a finite gap between ground state and the continuum of excited states at the collapse point. For large qubit spli…
Quantum multi--photon spin--boson model is considered. We solve an operator Riccati equation associated with that model and present a candidate for a generalized parity operator allowing to transform spin--boson Hamiltonian to a block diagonal form what indicates an existence of the related symmetry of the model.
We investigate the quantum phase transition in the dissipative anisotropic quantum Rabi model in the framework of quantum dressed master equation. From perspectives of both numerical and analytical analysis, we unravel the implication of the giant photon-bunching feature on the first-order quantum phase transition. The observed two-photon statistics can be well described analytically within a few lowest eigenstates at the low temperature. Moreover, such significant photon-bunching peak is generally exhibited at the deep-strong qubit-photon coupling, which is however lacking in the dissipative …
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 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…
A system evolving under the driven Jaynes--Cummings model will undergo a phase transition at a critical driving field amplitude. This transition is foreshadowed by a collapse of the quasienergy level spectra of the system and remains present as the model is extended to include a counter-rotating interaction. We study this critical response and obtain the eigenvalues and eigenstates of the extended model by presenting a correspondence between the Jaynes--Cummings model and a charged Dirac particle subject to an external electromagnetic field. Under this correspondence, the field and two-level s…
We demonstrate that four-wave mixing processes in high-quality non-linear resonators can lead to Rabi-like oscillations in photon occupation numbers and second-order correlation functions, being a characteristic feature of the presence of entangled photon pairs in the optical signal. In the case of a system driven by a continuous coherent pump, the oscillations occur in the transient regime. We show that driving the system with pulsed coherent pumping would generate strongly anti-bunched photon states.
The two-photon Rabi Hamiltonian is a simple model describing the interaction of light with matter, with the interaction being mediated by the exchange of two photons. Although this model is exactly soluble in the rotating-wave approximation, we work with the full Hamiltonian, maintaining the non-integrability of the model. We demonstrate that, despite this non-integrability, there exist isolated, exact solutions for this model analogous to the so-called Juddian solutions found for the single-photon Rabi Hamiltonian. In so doing we use a Bogoliubov transformation of the field mode, as described…
In this paper, we give an analytical treatment to study the behavior of the collapse and the revival of the Rabi oscillations in the Jaynes-Cummings model (JCM). The JCM is an exactly soluble quantum mechanical model, which describes the interaction between a two-level atom and a single cavity mode of the electromagnetic field. If we prepare the atom in the ground state and the cavity mode in a coherent state initially, the JCM causes the collapse and the revival of the Rabi oscillations many times in a complicated pattern in its time-evolution. In this phenomenon, the atomic population invers…
We study the Two Photon Quantum Rabi Model by way of its spectral functions and survival probabilities. This approach allows numerical precision with large truncation numbers, and thus exploration of the spectral collapse. We provide independent checks and calibration of the numerical results by studying an exactly solvable case and comparing the essential qualitative structure of the spectral functions. We stress that the large time limit of the survival probability provides us with an indicator of spectral collapse, and propose a technique for the detection of this signal in the current and …
We have numerically obtained theoretical results for the collapse and the revival of the quantum Rabi oscillations for low average number of coherent photons injected on a two-level system in a lossy resonant cavity. We have adopted the multimode Jaynes-Cummings model for the same and especially treated the ``Ohmic" loss to the walls of the cavity, the leakage from the cavity, and the loss due to the spontaneous emission through the open surface of the cavity. We have compared our results with the experimental data obtained by Brune et al [Phys. Rev. Lett. 76, 1800 (1996)] in this regard.
We propose a superconducting circuit to implement a two-photon quantum Rabi model in a solid-state device, where a qubit and a resonator are coupled by a two-photon interaction. We analyze the input-output relations for this circuit in the strong coupling regime and find that fundamental quantum optical phenomena are qualitatively modified. For instance, two-photon interactions are shown to yield single- or two-photon blockade when a pumping field is either applied to the cavity mode or to the qubit, respectively. In addition, we derive an effective Hamiltonian for perturbative ultrastrong two…
Tho-photon Rabi oscillations hold potential for quantum computing and quantum information processing, because during a Rabi cycle a pair of entangled photons may be created. We theoretically investigate the onset of this phenomenon in a heterodimer comprising a semiconductor quantum dot strongly coupled to a metal nanoparticle. Two-photon Rabi oscillations in this system occur due to a coherent two-photon process involving the ground-to-biexciton transition in the quantum dot. The presence of a metal nanoparticle nearby the quantum dot results in a self-action of the quantum dot via the metal …
Within the framework of exact quantum electrodynamics in dispersing and absorbing media, we have studied the emission from an initially in the upper state prepared emitter in a high quality cavity in the case, when there are two cavity modes that resonantly interact with the emitter. In particular, when one of the modes is in resonance with the emitter transition and the other mode is tuned to the frequency equal to the Rabi splitting of the first mode, the effect of Rabi resonance is observed.
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…
Quantum Criticality from Spectral Collapse in the Two-Photon Rabi Model
Institute for Advanced Study in Physics and School of Physics, Zhejiang University（浙江大学物理学院和高等研究院）；Zhejiang Key Laboratory of Micro-Nano Quantum Chips and Quantum Control, School of Physics, Zhejiang University（浙江大学物理学院微纳量子芯片与量子控制浙江省重点实验室）
The true level crossings in a subspace of the asymmetric two-photon quantum Rabi model (tpQRM) have been observed when the bias parameter of qubit is an even multiple of the renormalized cavity frequency. Generally, such level crossings imply the existence of the hidden symmetry because the bias term breaks the obvious symmetry exactly. In this work, we propose a Bogoliubov operator approach (BOA) for the asymmetric tpQRM to derive the symmetry operators associated with the hidden symmetry hierarchically. The explicit symmetry operators consisting of Lie algebra at low biases can be easily obt…
We propose a robust realization of the two-photon quantum Rabi model in a trapped-ion setting based on a continuous dynamical decoupling scheme. In this manner the magnetic dephasing noise, which is identified as the main obstacle to achieve long time coherent dynamics in ion-trap simulators, can be safely eliminated. More specifically, we investigate the ultrastrong coupling regime of the two-photon quantum Rabi model whose realization in trapped ions involves second-order sideband processes. Hence, the resulting dynamics becomes unavoidably slow and more exposed to magnetic noise requiring a…
A Rabi dimer is used to model a recently reported circuit quantum electrodynamics system composed of two coupled transmission-line resonators with each coupled to one qubit. In this study, a phonon bath is adopted to mimic the multimode micromechanical resonators and is coupled to the qubits in the Rabi dimer. The dynamical behavior of the composite system is studied by the Dirac-Frenkel time-dependent variational principle combined with the multiple Davydov D$_{2}$ ansätze. Initially all the photons are pumped into the left resonator, and the two qubits are in the down state coupled with the …
We discuss a cavity-QED scheme to deterministically generate entangled photons pairs by using a three-level atom successively coupled to two single longitudinal mode high-Q cavities presenting polarization degeneracy. The first cavity is prepared in a well defined Fock state with two photons with opposite circular polarizations while the second cavity remains in the vacuum state. A half-of-a-resonant Rabi oscillation in each cavity transfers one photon from the first to the second cavity, leaving the photons entangled in their polarization degree of freedom. The feasibility of this implementat…
Recent progress on qubit manipulation allows application of periodic driving signals on qubits. In this study, a harmonic driving field is added to a Rabi dimer to engineer photon and qubit dynamics in a circuit quantum electrodynamics device. To model environmental effects, qubits in the Rabi dimer are coupled to a phonon bath with a sub-Ohmic spectral density. A non-perturbative treatment, the Dirac-Frenkel time-dependent variational principle together with the multiple Davydov D$_2$ {\it Ansatz} is employed to explore the dynamical behavior of the tunable Rabi dimer. In the absence of the p…
In this paper, we uncover the elusive level crossings in a subspace of the asymmetric two-photon quantum Rabi model (tpQRM) when the bias parameter of qubit is an even multiple of the renormalized cavity frequency. Due to the absence of any explicit symmetry in the subspace, this double degeneracy implies the existence of the hidden symmetry. The non-degenerate exceptional points are also given completely. It is found that the number of the doubly degenerate crossing points in the asymmetric tpQRM is comparable to that in asymmetric one-photon QRM in terms of the same order of the constrained …
We have found the algebraic structure of the two-qubit quantum Rabi model behind the possibility of its novel quasi-exact solutions with finite photon numbers by analyzing the Hamiltonian in the photon number space. The quasi-exact eigenstates with at most $1$ photon exist in the whole qubit-photon coupling regime with constant eigenenergy equal to single photon energy \hbarω, which can be clear demonstrated from the Hamiltonian structure. With similar method, we find these special "dark states"-like eigenstates commonly exist for the two-qubit Jaynes-Cummings model, with $E=N\hbarω$ (N=-1,0,1…
We use a quasi Hamiltonian formalism to describe the dissipative dynamics of a circuit QED qubit that is affected by several fluctuating two level systems with a 1/f noise power spectrum. The qubit-resonator interactions are described by the Jaynes Cummings model. We argue that the presence of pure dephasing noise in such a qubit-resonator system will also induce an energy relaxation mechanism via a fluctuating dipole coupling term. This random modulation of the coupling is seen to lead to rich physical behavior. For non-Markovian noise, the coupling can either worsen or alleviate decoherence …
We develop a fully quantum theoretical approach which describes the dynamics of Frenkel excitons and bi-excitons induced by few photon quantum light in a quantum well or wire (atomic chain) of finite size. The eigenenergies and eigenfunctions of the coupled exciton-photon states in a multiatomic system are found and the role of spatial confinement as well as the energy quantization effects in 1D and 2D cases is analyzed. Due to the spatial quantization, the excitation process is found to consist in the Rabi-like oscillations between the collective symmetric states characterized by discrete ene…
We present a mean-photon-number dependent variational method, which works well in whole coupling regime if the photon energy is dominant over the spin-flipping, to evaluate the properties of the Rabi model for both the ground state and the excited states. For the ground state, it is shown that the previous approximate methods, the generalized rotating-wave approximation (only working well in the strong coupling limit) and the generalized variational method (only working well in the weak coupling limit), can be recovered in the corresponding coupling limits. The key point of our method is to ta…
In this report, we have studied exceptional spectra of the two-qubit quantum Rabi model in two situations. Firstly, an exceptional spectra is achieved in resonant condition, in which the frequencies of two qubit and photon field satisfy resonant relation. With transformed rotating-wave approximation (TRWA) method, the Rabi model can be mapped into the solvable formation in qubit-photon Fock states space. Based on the quantum supperconducting circuits experiment setup, the best value range of the system parameters are discussed. In the resonant station working window, the energy spectrums are c…
We employ a polaron picture to investigate the properties of the two-photon quantum Rabi model (QRM), which describes a two-level or spin-half system coupled with a single bosonic mode by a two-photon process. In the polaron picture, the coupling in the two-photon process leads to spin-related asymmetry so that the original single bosonic mode splits into two separated frequency modes for the opposite spins, which correspond to two \textit{bare} polarons. Furthermore, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional \textit{i…
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…
Isolation of a single atomic particle and monitoring its resonance fluorescence is a powerful tool for studies of quantum effects in radiation-matter interaction. Here we present observations of quantum dynamics of an isolated neutral atom stored in a magneto-optical trap. By means of photon correlations in the atom's resonance fluorescence we demonstrate the well-known phenomenon of photon antibunching which corresponds to transient Rabi oscillations in the atom. Through polarization-sensitive photon correlations we show a novel example of resolved quantum fluctuations: spontaneous magnetic o…
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 …
A coherent coupling among different energy photons provided by nonlinear optical interaction is regarded as a photonic version of the Rabi oscillation. Cavity enhancement of the nonlinearity reduces energy requirement significantly and pushes the scalability of the frequency-encoded photonic circuit based on the photonic Rabi oscillation. However, confinement of the photons in the cavity severely limits the number of interactable frequency modes. Here we demonstrate a wide-bandwidth and efficient photonic Rabi oscillation achieving full-cycle oscillation based on a cavity-enhanced nonlinear op…
The analytical exact solutions to the mixed quantum Rabi model (QRM) including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of $4 \times 4$ determinants responsible for the exact solutions are derived. These so-called $G$-functions with pole structures can be reduced to the previous ones in the unmixed QRMs. The zeros of $G$-functions reproduce completely the regular spectra. The exceptional eigenvalues can also be obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photo…
In this work, we investigate the quantum phase transition of light in the dissipative Rabi-Hubbard lattice under the framework of the mean-field theory and quantum dressed master equation. The order parameter of photons in strong qubit-photon coupling regime is derived analytically both at zero and low temperatures. Interestingly, we can locate the localization and delocalization phase transition very well in a wide parameter region. {In particular for the zero-temperature limit, the critical tunneling strength approaches zero generally in the deep-strong qubit-photon coupling regime, regardle…
In a hybrid optomechanical setup consisting of a two-level atom in a cavity with a pendular end-mirror, the interplay between the light field's radiation pressure on the mirror and the dipole interaction with the atom can lead to a blockade effect, which manifests itself in a suppression of Rabi oscillations in the atomic population. This effect is present when the system is in the single photon strong coupling regime and has an analogy in the photon blockade of optomechanics.
We realize a photonic analog simulator of the quantum Rabi model, based on light transport in femtosecond-laser-written waveguide superlattices, which provides an experimentally accessible testbed to explore the physics of light-matter interaction in the deep strong coupling regime. Our optical setting enables to visualize dynamical regimes not yet accessible in cavity or circuit quantum electrodynamics, such as bouncing of photon number wave packets in parity chains of Hilbert space.
We determine the Rabi oscillations for coherent and squeezed coherent states via one- and two-photon atomic transitions in two- and three-level atoms, respectively. The effect of squeezing coherent states is to reduce the large number of photons of mesoscopic scale to fewer photons and so be able to use numerical solutions with the aid of the Jaynes-Cummings model. We maximize the squeezing of the coherent state by minimizing the ratio of the photon number variance to photon number, thus narrowing the region where revival occurs.
Two-photon Rabi splitting in a cavity-dot system provides a basis for multi-qubit coherent control in quantum photonic network. Here we report on two-photon Rabi splitting in a strongly coupled cavity-dot system. The quantum dot was grown intentionally large in size for large oscillation strength and small biexciton binding energy. Both exciton and biexciton transitions couple to a high quality factor photonic crystal cavity with large coupling strengths over 130 $μ$eV. Furthermore, the small binding energy enables the cavity to simultaneously couple with two exciton states. Thereby two-photon…
