We studied the two-qubit quantum Rabi model and found its dark state solutions with at most N photons. One peculiar case presents when $N=3$, which has constant eigenenergy in the whole coupling regime and leads to level crossings within the same parity subspace. We also discovered asymptotic solutions with at most $N=2i+3$ $(i=1,2,3,\dots)$ photons, and constant eigenenergy $N\hbar ω$ when coupling $g$ becomes much larger than photon frequency $ω$. Although generally all photon number states are involved in the two-qubit quantum Rabi model, such $N$-photon solutions exist and may have applications in quantum information processing with ultrastrong couplings.
Using extended coherent states, an analytically exact study has been carried out for the quantum Rabi model with two equivalent qubits. Compact transcendental functions of one variable have been derived leading to exact solutions. The energy spectrum is clearly identified and analyzed. Also obtained analytically is the necessary and sufficient conditions for the occurrence of isolated exceptional solutions, which are not doubly degenerate as in the one-qubit quantum Rabi model.
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…
A two-interacting-qubit quantum Rabi-like model with vanishing transverse fields on the qubit-pair is studied. Independently of the coupling regime, this model can be exactly and unitarily reduced to two independent single-spin quantum Rabi models, where the spin-spin coupling plays the role of the transverse field. This transformation and the analytical treatment of the single-spin quantum Rabi model provide the key to prove the integrability of our model. The existence of different first-order quantum phase transitions, characterized by discontinuous two-spin magnetization, mean photon numbe…
For a system composed of nine qubits, we show that natural interactions among the qubits induce the time evolution that can be regarded, at discrete times, as the Rabi oscillation of a logical qubit. Neither fine tuning of the parameters nor switching of the interactions is necessary. Although straightforward application of quantum error correction fails, we propose a protocol by which the logical Rabi oscillation is protected against all single-qubit errors. The present method thus opens a simple and realistic way of protecting the unitary time evolution against noise.
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 study the ground states of the single- and two-qubit asymmetric Rabi models, in which the qubit-oscillator coupling strengths for the counterrotating-wave and corotating-wave interactions are unequal. We take the transformation method to obtain the approximately analytical ground states for both models and numerically verify its validity for a wide range of parameters under the near-resonance condition. We find that the ground-state energy in either the single- or two-qubit asymmetric Rabi model has an approximately quadratic dependence on the coupling strengths stemming from different cont…
All existing quantum oblivious transfer protocols are to realize the oblivious transfer of bit or bit-string. In this paper, p-Rabin quantum oblivious transfer of a qubit (abbr. p-Rabin qubit-OT) is achieved by using a probabilistic teleportation (abbr. PT) protocol. Here, this is the first time that the concept of qubit-OT is presented. As the PT protocol is able to transfer an (un)known pure state with a certain probability, this feature makes the PT protocol well fit for Rabin OT. Furthermore, the PT protocol can be used for OT of a bit by encoding classical bit with two pre-agreed orthogon…
Multi-qubit quantum Rabi model, which is a fundamental model describing light-matter interaction, plays an important role in various physical systems. In this paper, we propose a theoretical method to simulate multi-qubit quantum Rabi model in a circuit quantum electrodynamics system. By means of external transversal and longitudinal driving fields, an effective Hamiltonian describing the multi-qubit quantum Rabi model is derived. The effective frequency of the resonator and the effective splitting of the qubits depend on the external driving fields. By adjusting the frequencies and the amplit…
We proposed a scheme to demonstrate macroscopic quantum jumps in a superconducting phase qubit coupled to a microscopic two-level system in the Josephson tunnel junction. Irradiated with suitable microwaves, the Rabi oscillations of the qubit exhibit signatures of quantum jumps: a random telegraph signal with long intervals of intense macroscopic quantum tunneling events (bright periods) interrupted by the complete absence of tunneling events (dark periods). An analytical model was developed to describe the width of the dark periods quantitatively. The numerical simulations indicate that our a…
We consider a generalized Rabi model formed by two identical qubits interacting with a common oscillator mode. In the near resonance configuration where the oscillator frequency is close to the transition frequency of the qubit, we determine the ground state of the model approximately by using a variational method and a transformation method. Both methods are shown to provide good agreement with the exact numerical ground state for a range of coupling strength in the ultrastrong regime. In addition, we indicate how the accuracy of the approximation can be further improved by using perturbation…
The polaron picture is employed to investigate and elucidate the physics of the two-qubit quantum Rabi model, which describes two identical qubits coupled to a common harmonic oscillator. This approach enables us to obtain the ground-state energy and some other simpler physical observables with high accuracy in all regimes of the coupling strengths $g$, while there is no constraint to the ration of tunneling frequency $Ω$ and field frequency $ω$, which is not simultaneously possible using previous methods. Besides, we also discover a new phenomenon which is not present in the one-qubit Rabi mo…
Using extended coherent states, an analytical exact study has been carried out for the quantum Rabi model (QRM) with two arbitrary qubits in a very concise way. The $G$-functions with $2 \times 2$ determinants are generally derived. For the same coupling constants, the simplest $G$-function, resembling that in the one-qubit QRM, can be obtained. Zeros of the $G$-function yield the whole regular spectrum. The exceptional eigenvalues, which do not belong to the zeros of the $G$ function, are obtained in the closed form. The Dark states in the case of the same coupling can be detected clearly in …
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…
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…
We examine the influence of environmental interactions on simple quantum systems by obtaining the exact reduced dynamics of a qubit coupled to a one-dimensional spin bath. In contrast to previous studies, both the qubit-bath coupling and the nearest neighbor intrabath couplings are taken as the spin-flip XX-type. We first study the Rabi oscillations of a single qubit with the spin bath prepared in a spin coherent state, finding that nonresonance and finite intrabath interactions have significant effects on the qubit dynamics. Next, we discuss the bath-induced decoherence of the qubit when the …
Jaynes-Cummings dynamics in strong coupling for many-interacting-qubit quantum Rabi models
University of Catania（卡塔尼亚大学）；INFN Sez. Catania（意大利国家核物理研究所卡塔尼亚分部）；Università degli Studi di Palermo（巴勒莫大学）；CNR-IMM, UoS Università（意大利国家研究委员会微电子与微系统工程研究所卡塔尼亚大学联合实验室）
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…
In nanomechanical QED system, consisting of a charge qubit and a nonlinear nanomechanical resonator, we study the temporal behavior of Rabi oscillation in the nonlinear Jaynes-Cummings model. Using microscopic master equation approach, we solve time evolution of the density operator describing this model. Also, the probability of excited state of charge qubit is calculated. These analytic calculations show how nonlinearity parameter and decay rates of two different excited states of the qubit-resonator system affect time-oscillating and decaying of Rabi oscillation.
We study the two-qubit Rabi model in the most general case where the qubits are different from each other. The spectrum of the system in the ultrastrong-coupling regime is shown to converge to two forced oscillator chains by perturbation theory. An even and odd decomposition of the Hilbert space allows us to calculate the spectra in any given parameter regime; the cases studied confirm our perturbation theory prediction in the ultrastrong-coupling regime and point to crossings in the spectra within each parity subspace in the moderate-coupling regime. The normal modes of the system are calcula…
We present a physically motivated variational wave function for the ground state of the asymmetric quantum Rabi model (AQRM). The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states, and relies only on three variational parameters in the regimes of interest where the squeezing effect becomes negligible. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. We use the variational result to calculate various relevant physical observables of the ground state…
Two-qubit system is the foundation of constructing the universal quantum gate. We have studied the two-qubit Rabi model for the general case and its generalizations with dipole, XXX and XYZ Heisenberg qubit-qubit interactions, which are commonly used in quantum computation. Their solutions are presented analytically with eigenstates can be obtained in terms of extended coherent states or Fock states and applied to the construction of the ultrafast two-qubit quantum gate in circuit QED and quantum state storage and transfer. Some novel kinds of quasi-exact solutions are found for specific sets …
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 …
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…
This paper provides a scheme for generating maximally entangled qubit states in the anti-Jaynes-Cummings interaction mechanism, so called entangled anti-polariton qubit states. We demonstrate that in an initial vacuum-field, Rabi oscillations in a cavity mode in the anti-Jaynes-Cummings interaction process, occur in the reverse sense relative to the Jaynes-Cummings interaction process and that time evolution of entanglement in the anti-Jaynes-Cummings interaction process takes the same form as in the Jaynes-Cummings interaction process. With the generated anti-polariton qubit state as one of t…
We analyze the dynamics of the quantum Rabi model for two qubits interacting through a common bosonic field mode (resonator), focusing on the generation and detection of maximally entangled Bell states. We obtain analytical results for the unitary dynamics of this system in the slow-qubit (or degenerate) regime, considering ultra-strong coupling between qubits and resonator mode, for which the rotating wave approximation is no longer applicable. We also numerically investigate the dynamics beyond the slow-qubit condition in order to study the validity of the model in the presence of less stric…
We demonstrate theoretically the noise-stimulated enhancement of quantum coherence in a superconducting flux qubit. First, an external classical noise can increase the off-diagonal components of the qubit density matrix. Second, in the presence of noise, the Rabi oscillations survive for times significantly longer than the Rabi decay time in a noiseless system. These Rabi oscillations appear as a modulation of the forced response of the qubit to the ac driving field. These effects can be considered as a manifestation of quantum stochastic resonance and are relevant to experimental techniques, …
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 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 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…
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…
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 diago…
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…
For a superconducting qubit driven to perform Rabi oscillations and coupled to a slow electromagnetic or nano-mechanical oscillator we describe previously unexplored quantum optics effects. When the Rabi frequency is tuned to resonance with the oscillator the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the qubit and a bi-stability with lasing behavior of the oscillator; for red detuning the qubit cools the oscillator. This behavior persists at the symmetry point where the qubit-oscillator coupling is quadratic and decoherence effects are m…
Near-optimal synthesis of non-Gaussian phase gates via qubit-oscillator Rabi control
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学基础与前沿研究院）；Yangtze Delta Industrial Innovation Center of Quantum Science and Technology（长三角量子科技产业创新中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Exploring highly connected networks of qubits is invaluable for implementing various quantum algorithms and simulations as it allows for entangling qubits with reduced circuit depth. Here, we demonstrate a multi-qubit STAR (Sideband Tone Assisted Rabi driven) gate. Our scheme is inspired by the ion qubit Mølmer-Sørensen gate and is mediated by a shared photonic mode and Rabi-driven superconducting qubits, which relaxes restrictions on qubit frequencies during fabrication and supports scalability. We achieve a two-qubit gate with maximum state fidelity of 0.95 in 310 ns, a three-qubit gate with…
We present theoretical transmission spectra of a strongly driven, damped, flux qubit coupled to a dissipative resonator in the ultrastrong coupling regime. Such a qubit-oscillator system, described within a dissipative Rabi model, constitutes the building block of superconducting circuit QED platforms. The addition of a strong drive allows one to characterize the system properties and study novel phenomena, leading to a better understanding and control of the qubit-oscillator system. In this work, the calculated transmission of a weak probe field quantifies the response of the qubit, in freque…
A quantum two level system coupled to a harmonic oscillator represents a ubiquitous physical system. New experiments in circuit QED and nano-electromechanical systems (NEMS) achieve unprecedented coupling strength at large detuning between qubit and oscillator, thus requiring a theoretical treatment beyond the Jaynes Cummings model. Here we present a new method for describing the qubit dynamics in this regime, based on an oscillator correlation function expansion of a non-Markovian master equation in the polaron frame. Our technique yields a new numerical method as well as a succinct approxima…
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…
We investigate a superconducting circuit consisting of multiple capacitively-coupled charge qubits. The collective Rabi oscillation of qubits is numerically studied in detail by imitating environmental fluctuations according to the experimental measurement. For the quantum circuit composed of identical qubits, the energy relaxation of the system strongly depends on the interqubit coupling strength. As the qubit-qubit interaction is increased, the system's relaxation rate is enhanced firstly and then significantly reduced. In contrast, the inevitable inhomogeneity caused by the nonideal fabrica…
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…
The dispersive regime of circuit QED is the main workhorse for todays quantum computing prototypes based on superconducting qubits. Analytic descriptions of this model typically rely on the rotating wave approximation of the interaction between the qubits and resonators, using the Jaynes-Cummings model as starting point for the dispersive transformation. Here we present analytic results on the dispersive regime of the dissipative Rabi model, without taking the rotating wave approximation of the underlying Hamiltonian. Using a recently developed hybrid perturbation theory based on the expansion…
