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arXiv:2309.00729 · SciPost Physics

Invariant approach to the driven Jaynes-Cummings model

Ivan A. Bocanegra-Garay, Leonardi Hernández-Sánchez, I. Ramos-Prieto, Francisco Soto‐Eguibar, H. M. Moya-Cessa

Published Jan 8, 2024• 15 citations
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Abstract

We investigate the dynamics of the driven Jaynes-Cummings model, where a two-level atom interacts with a quantized field and both, atom and field, are driven by an external classical field. Via an invariant approach, we are able to transform the corresponding Hamiltonian into the one of the standard Jaynes-Cummings model. Subsequently, the exact analytical solution of the Schrödinger equation for the driven system is obtained and employed to analyze some of its dynamical variables.

DOI
10.21468/scipostphys.16.1.007

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We investigate the dynamics of the driven Jaynes-Cummings model, where a two-level atom interacts with a quantized field and both, atom and field, are driven by an external classical field. Via an invariant approach, we are able to transform the corresponding Hamiltonian into the one of the standard Jaynes-Cummings model. Subsequently, the exact analytical solution of the Schrödinger equation for the driven system is obtained and employed to analyze some of its dynamical variables.
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.
We study the dynamical features of the Jaynes–Cummings model with the atom driven by an external classical field in the case that the cavity field is initially in a Fock state.We find the dynamical variables by using dressed states in the bare atom representation. We show that the dynamical behavior of the cavity field can be strongly modified by the external classical field. We also find that the cavity field is super-Poissonian after some interaction time if the external field is strong enough.
Abstract This study explores the dynamics of a Driven Time‐Dependent Jaynes–Cummings (DTDJC) model, in which the atom and the field are influenced by an external classical field with time‐varying amplitude modulation. By applying a series of unitary transformations and establishing suitable conditions for the time‐dependent parameters of the system, the Hamiltonian operator is transformed into one representing a system not driven by an external field, namely a Free TDJC (FTDJC) system. This mapping allows us to determine, in both the time‐independent and time‐dependent cases, forms of the pumping modulating the field and the atomic population that are amenable to exact analytical solution. The effects of an off‐resonance parameter emerging from the analytical solution of the model and the intensity of the classical pumping field on the dynamics of the atomic population are investigated. The exact solution allows us to reveal a rich dynamics of the system (including long‐time super‐revivals) that could be of use in future quantum technologies.
Collapse and revivals in the Jaynes-Cummings model are studied within the context of the Mollow transformation for a single-mode coherent-state cavity field. The Mollow-transformed Jaynes-Cummings Hamiltonian has two atom-field terms, one corresponding to a classical field and the other to a quantized field driving the atomic transition. As such it maps onto the complementary problem of an atom in an optical cavity, whose field is initially in the vacuum state, driven by an external classical field. Both problems have the same atomic-state dynamics. It is shown that the revivals can be associated with two distinct properties of the quantized field of the transformed Hamiltonian. Revivals occurring at even (odd) multiples of the revival time are correlated with field states that are close to the initial field state and for which the average energy in the field is a minimum (maximum). Using semiclassical dressed states, we are able to map the problem for the field states onto one involving two uncoupled oscillators driven by off-resonant fields, provided effects related to dispersion are neglected. Analytic expressions for the photon number distribution of the transformed cavity field, including dispersion, are obtained for the even revival times.