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arXiv:1808.06727 · Physica Scripta

Quasienergy collapse in the driven Jaynes–Cummings–Rabi model: correspondence with a charged Dirac particle in an electromagnetic field

R. Gutiérrez-Jáuregui, H. J. Carmichael

Published Jul 31, 2018• 11 citations
arXivHTMLPDF

Abstract

A system evolving under the driven Jaynes–Cummings (JC) 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 JC model and a charged Dirac particle subject to an external electromagnetic field. Under this correspondence, the field and two-level system that compose the JC model map to the external and internal degrees of freedom describing the Dirac particle, respectively. The phases of the system below (above) the critical drive are then characterized by discrete (continuous) solutions, with the manipulations required to obtain these solutions appearing naturally as Lorentz transformations.

DOI
10.1088/1402-4896/aad6fc

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A system evolving under the driven Jaynes–Cummings (JC) 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 JC model and a charged Dirac particle subject to an external electromagnetic field. Under this correspondence, the field and two-level system that compose the JC model map to the external and internal degrees of freedom describing the Dirac particle, respectively. The phases of the system below (above) the critical drive are then characterized by discrete (continuous) solutions, with the manipulations required to obtain these solutions appearing naturally as Lorentz transformations.
When the photonic mode in the Jaynes-Cummings model is driven by an external classical field, the system can undergo the photon-blockade breakdown phase transition at a critical point. Such a phase transition has been detailedly investigated, but the critical properties of the eigenstates remain largely unexplored so far. We here study the geometric criticality associated with these eigenstates. The amplitude and phase of the drive serve as the control parameter of the governing Hamiltonian. We find the quantum metric and Berry curvature tensors for each eigenstate display divergent behaviors in the critical region. More importantly, the divergence associated with bright eigenstates is much more pronounced than that for the unique dark state. Our theoretical results can be experimentally confirmed in circuit quantum electrodynamics systems, where the driven Jaynes-Cummings model has been realized.
Abstract 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 specied by a conserved JC excitation number operator which generates the U(1) symmetry of RF and a correlated counter-rotating frame (CRF) dominated by an exactly solved antiJaynes-Cummings (AJC) interaction specied by a conserved AJC excitation number operator which generates the U(1) symmetry of CRF (c) for QRM dynamical evolution in RF, the initial atom-eld state je0i is an eigenstate of the effective AJC Hamiltonian HAJC, while the effective JC Hamiltonian HJC drives this initial state je0i into a time evolving entangled state, and, in a corresponding process for QRM dynamical evolution in CRF, the initial atom-eld state jg0i is an eigenstate of the effective JC Hamiltonian, while the effective AJC Hamiltonian drives this initial state jg0i into a time evolving entangled state, thus addressing one of the long-standing challenges of theoretical and experimental QRM dynamics; consistent generalizations of the initial states je0i , jg0i to corresponding n 0 entangled eigenstates j+en i , j 􀀀g ni of the AJC in RF and JC in CRF, respectively, provides general dynamical evolution of QRM characterized by collapses and revivals in the time evolution of the atomic, eld mode, JC and AJC excitation numbers for large initial photon numbers ; the JC and AJC excitation numbers are conserved in the respective frames RF, CRF, but each evolves with time in the alternate frame.
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 analyze the time-dependent physical spectrum of a driven Jaynes-Cummings model in which both the two-level system and the quantized cavity mode are subject to coherent classical driving. The time-dependent Hamiltonian is mapped, via well-defined unitary transformations, onto an effective stationary Jaynes-Cummings form. Within this framework, we derive closed-form expressions for the two-time correlation functions of both the atomic and field operators. These correlation functions are subsequently used to evaluate the time-dependent physical spectrum according to the Eberly-Wódkiewicz definition, which properly accounts for finite spectral resolution and transient emission dynamics. We show that the external driving leads to substantial modifications of the atomic spectral response, including controllable frequency shifts and asymmetric line shapes. Importantly, we identify a regime in which the driving parameters are chosen such that the coherent displacement induced in the cavity field exactly cancels out the initial coherent amplitude. In this limit, the system dynamics reduce to that of an effectively vacuum-initialized Jaynes-Cummings model, and the standard vacuum Rabi splitting is recovered. This behavior provides a clear and physically transparent interpretation of the spectral features as arising from coherent field displacement rather than from modifications of the underlying atom-cavity coupling.
A two-level system driven by a classical single frequency field in rotating-wave approximation is a basic model for resonance in quantum optics. The probability of making a transition depends sinusoidally on time with a characteristic frequency called the Rabi frequency. The Jaynes-Cummings model (JCM) consists of a two-level atomic system driven by a single quantized electromagnetic field mode. The field entropy in the case of an atom interacting with an initially coherent field is expected to reflect the collapse and revival of Rabi oscillations. The photon statistics of the JCM are examined and the entropy concepts are used to describe the apparently disordered collapses. A density matrix method is used to describe the influence of photon statistics in the JCM. The quantum generalization defines entropy through where the trace is taken over a convenient basis set. The thermal field JCM shows enhanced photon variances as expected for an initially Bose-Einstein field with large fluctuations.