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Physical Review A

Response of a two-level atom to a classical field and a quantized cavity field of different frequencies

C. K. Law, Joseph H. Eberly

Published Jun 1, 1991• 29 citations
Publisher

Abstract

We consider a two-level atom interacting simultaneously with a classically described cw laser field and a quantized cavity field. The classical field is assumed strong, and the cavity field is regarded as a quantum probe of its dressed states. The two fields have different frequencies and amplitudes, but we show that an effective time-independent Hamiltonian can be obtained via two rotating-frame transformations. There is a different effective Hamiltonian in each of an infinite sequence of ``probe resonance zones'' associated with the subharmonic Rabi sidebands of the dressed states. In the mth zone, the Hamiltonian has the same form as the m-photon Jaynes-Cummings Hamiltonian. We consider both time-dependent and time-averaged atomic response and make comparisons with recent findings in the case that both fields are fully classical.

DOI
10.1103/physreva.43.6337

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We consider a two-level atom interacting simultaneously with a classically described cw laser field and a quantized cavity field. The classical field is assumed strong, and the cavity field is regarded as a quantum probe of its dressed states. The two fields have different frequencies and amplitudes, but we show that an effective time-independent Hamiltonian can be obtained via two rotating-frame transformations. There is a different effective Hamiltonian in each of an infinite sequence of ``probe resonance zones'' associated with the subharmonic Rabi sidebands of the dressed states. In the mth zone, the Hamiltonian has the same form as the m-photon Jaynes-Cummings Hamiltonian. We consider both time-dependent and time-averaged atomic response and make comparisons with recent findings in the case that both fields are fully classical.
We consider a two-channel \ensuremath{\Lambda}-type three-level atom interacting simultaneously with two quantized cavity fields and a classically described cw laser field, all of different frequencies. The effective interactions are of the Raman type, in which the two cavity fields are regarded as Stokes and anti-Stokes modes, and the classical field acts as the pump mode, so that both the pump-Stokes and the pump--anti-Stokes transitions are included. We present the analytic solution of the system evolution operator. We also study the time dependence of the atomic inversion and the statistical properties of the cavity fields. We show that strongly correlated photons can be generated in our model.
We present a general procedure to describe the dynamics of N degenerate quantized fields interacting resonantly with a two–level atom, all coupled with the same strength, within the rotating–wave approximation. Starting from the analysis of the two and three field cases, we generalize the method by identifying dynamical invariants that lead to a factorized form of the time–evolution operator. A unitary transformation reduces the problem to an effective Jaynes–Cummings Hamiltonian, where only one field interacts with the atom and the remaining modes contribute as free fields. Assuming initially coherent fields and an atomic superposition, we compute the atomic inversion and the mean photon number, revealing vacuum Rabi oscillations with a frequency determined by an effective coupling constant that exceeds the individual atom–field coupling, as well as the characteristic collapse-revival behavior.
We consider a \ensuremath{\Lambda}-type three-level atom interacting simultaneously with two quantized cavity fields and a classical field, all of different frequencies. The classical field is of constant amplitude and the quantized cavity fields are dynamical variables. The effective interactions are Raman type with the two cavity fields regarded as Stokes and anti-Stokes modes. The classical field acts as the pump-mode, so that both the pump--Stokes and the pump--anti-Stokes transitions are included. We present analytic expressions for the dressed states of the system in a general situation where the two channels have different coupling strengths. The dressed states are closely related to the two-mode squeezed number states. We find that two-mode squeezing and nontrivial phase correlation occur in the model. The effects of weak cavity damping are also discussed.
We study the two-photon interaction of a two-level atom with a quantized bimodal field of a loss-free cavity. We assume that both the coupling strength and the atom-field detuning may depend on the populations of the two modes. We show that this general nonlinear model is exactly solvable in terms of a dressing unitary operator. Exact expressions for the eigenstates as well as for the Rabi frequencies of the coupled system are given as explicit functions of the nonlinearity terms present in the model. Exploiting the flexibility of these results, we investigate the atomic inversion supposing that at t=0 one mode is coherent, the other is in a Fock state and the atom is excited. We compare the dynamical behaviour of various particular nonlinear models. A common feature is the presence of collapses and revivals of the Rabi oscillations. We find that the atomic inversion manifests exact periodicity when only one specific type of intensity-dependent term (coupling strength or detuning) is present. Such behaviour is physically interpreted in the framework of the dressed atom representation. The simulataneous presence of both nonlinear contributions in the model breaks such regular behaviour in the long-time atomic evolution, in which case competition effects occur which qualitatively modify the quantum dynamics of the system. Other quantum manifestations are brought to light. Their relevance, in connection with the possibility of preparing a cavity mode in an arbitrary Fock state, is briefly discussed.
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 consider a two-level atom in a strong laser field, described by the Jaynes-Cummings Hamiltonian in the rotating-wave approximation, yielding the well known dressed states. Diagonalization of the total Hamiltonian is attained by introducing dressed-atomic and photon operators. The quantum theory of measurement of finite duration is used to explore the nature of the physical observables corresponding to the dressed operators. A measuring apparatus is contrived in order to observe the bare-atomic population and the bare-photon number in a dressed state of the system. It is shown that, if the duration of the measurement is longer than the Rabi period of the system, the apparatus leads to observation of the dressed-atomic population and of the dressed-photon number rather than of the corresponding bare quantities for which it was designed. Conceptual and practical consequences of this result are discussed.