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arXiv:2003.09803 · Physical Review A

Jaynes-Cummings model under monochromatic driving

Leonardo Ermann, Gabriel G. Carlo, Alexei D. Chepelianskii, Dima L. Shepelyansky

Published Sep 30, 2020• 9 citations
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Abstract

We study analytically and numerically the properties of the Jaynes-Cummings model under monochromatic driving. The analytical results allow us to understand the regime of two branches of multiphoton excitation in the case of close resonance between resonator and driven frequencies. The rotating wave approximation allows us to reduce the description of the original driven model to an effective Jaynes-Cummings model with strong coupling between photons and qubit. The analytical results are in good agreement with the numerical ones even if there are certain deviations between the theory and numerics in the close vicinity of the resonance. We argue that the rich properties of the driven Jaynes-Cummings model represent a new area for experimental investigations with superconducting qubits and other systems.

DOI
10.1103/physreva.102.033729

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We study analytically and numerically the properties of the Jaynes-Cummings model under monochromatic driving. The analytical results allow us to understand the regime of two branches of multiphoton excitation in the case of close resonance between resonator and driven frequencies. The rotating wave approximation allows us to reduce the description of the original driven model to an effective Jaynes-Cummings model with strong coupling between photons and qubit. The analytical results are in good agreement with the numerical ones even if there are certain deviations between the theory and numerics in the close vicinity of the resonance. We argue that the rich properties of the driven Jaynes-Cummings model represent a new area for experimental investigations with superconducting qubits and other systems.
The Jaynes–Cummings model with or without rotating-wave approximation plays a major role to study the interaction between atom and light. We investigate the Jaynes–Cummings model beyond the rotating-wave approximation. Treating the counter-rotating terms as periodic drivings, we solve the model in the extended Floquet space. It is found that the full energy spectrum folded in the quasi-energy bands can be described by an effective Hamiltonian derived in the high-frequency regime. In contrast to the Z 2 symmetry of the original model, the effective Hamiltonian bears an enlarged U(1) symmetry with a unique photon-dependent atom-light detuning and coupling strength. We further analyze the energy spectrum, eigenstate fidelity and mean photon number of the resultant polaritons, which are shown to be in accordance with the numerical simulations in the extended Floquet space up to an ultra-strong coupling regime and are not altered significantly for a finite atom-light detuning. Our results suggest that the effective model provides a good starting point to investigate the rich physics brought by counter-rotating terms in the frame of Floquet theory.
We present the analytical solution of the Tavis-Cummings (TC) model for more than one qubit inhomogeneously coupled to a single mode radiation field beyond the rotating-wave approximation (RWA). The significant advantage of the displaced oscillator basis enables us to apply the same truncation techniques adopted in the single qubit Jaynes-Cummings (JC) model to the multiple qubits system. The derived analytical spectrum match perfectly the exact diagonalization numerical solutions of the inhomogeneous TC model in the parameter regime where the qubits transition frequencies are far off-resonance with the field frequency and the interaction strengths reach the ultra-strong coupling regime. The two-qubit TC model is quasi-exactly solvable because part of the spectra can be determined exactly in the homogeneous coupling case with two identical qubits or with symmetric(asymmetric) detuning. By means of the fidelity of quantum states we identify several nontrivial level crossing points in the same parity subspace, which implies that homogeneous coupled two-qubit TC model with ω1=ω2ω_1=ω_2ω1​=ω2​ or ω1±ω2=2ωcω_1\pmω_2=2ω_cω1​±ω2​=2ωc​ is integrable. We further explore the time evolution of the qubit's population inversion and the entanglement behavior taking two qubits as an example. The analytical methods provide unexpectedly accurate results in describing the dynamics of the qubit in the present experimentally accessible coupling regime, showing that the collapse-revival phenomena emerge, survive, and are finally destroyed when the coupling strength increases beyond the ultra-strong coupling regime. The suggested procedure applies readily to the multiple qubits system such as the GHZ state entanglement evolution and quantum entanglement between a single photon and superconducting qubits of particular experiment interest.
We analyze the driven resonantly coupled Jaynes-Cummings model in terms of a quasienergy approach by switching to a frame rotating with the external modulation frequency and by using the dressed atom picture. A quasienergy surface in phase space emerges whose level spacing is governed by a rescaled effective Planck constant. Moreover, the well-known multiphoton transitions can be reinterpreted as resonant tunneling transitions from the local maximum of the quasienergy surface. Most importantly, the driving defines a quasienergy well which is nonperturbative in nature. The quantum-mechanical quasienergy state localized at its bottom is squeezed. In the Purcell-limited regime, the potential well is metastable and the effective local temperature close to its minimum is uniquely determined by the squeezing factor. The activation occurs in this case via dressed spin-flip transitions rather than via quantum activation as in other driven nonlinear quantum systems such as the quantum Duffing oscillator. The local maximum is, in general, stable. However, in presence of resonant coherent or dissipative tunneling transitions the system can escape from it and a stationary state arises as a statistical mixture of quasienergy states being localized in the two basins of attraction. This gives rise to a resonant or an antiresonant nonlinear response of the cavity at multiphoton transitions. The model finds direct application in recent experiments with a driven superconducting circuit quantum electrodynamics setup.
We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the qubit-oscillator coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive qubit readout is possible. If several qubits are coupled to one resonator, an effective qubit-qubit interaction of Ising type emerges, whereas RWA leads to isotropic XYXYXY interaction. This impacts on the entanglement characteristics of the qubits.
Recent measurement on an LC resonator magnetically coupled to a superconducting qubit[arXiv:1005.1559] shows that the system operates in the ultra-strong coupling regime and crosses the limit of validity for the rotating-wave approximation of the Jaynes-Cummings model. By using extended bosonic coherent states, we solve the Jaynes-Cummings model exactly without the rotating-wave approximation. Our numerically exact results for the spectrum of the flux qubit coupled to the LC resonator are fully consistent with the experimental observations. The smallest Bloch-Siegert shift obtained is consistent with that observed in this experiment. In addition, the Bloch-Siegert shifts in arbitrary level transitions and for arbitrary coupling constants are predicted.
In this paper we define, in the context of dynamical algebras, a set of operators that are suitable for studying any relevant quantity related to the two-level Jaynes–Cummings model (JCM). We study the usual JCM with and without the rotating wave approximation (RWA), and then we add the presence of an external field. In this last case we find that in the strong driving regime there are significant differences between the results obtained with the RWA and without it. This holds even for the case in which the usual requirements for applying the RWA without an external driving are fulfilled.