[Paper Review] Stimulated Emission of Radiation in a Single Mode for both Resonance and Non-resonance for Various Initial Photon Distributions

M. T. Tavis, F.W. Cummings|arXiv (Cornell University)|Dec 16, 2012

Spectroscopy and Quantum Chemical Studies3 citations

TL;DR

This paper extends the quantum theory of stimulated emission in a single-mode cavity by analyzing both resonant and non-resonant interactions between a two-level atom and electromagnetic fields with diverse initial photon distributions—such as coherent, thermal, squeezed, and superposed coherent states. It demonstrates that non-resonant driving induces oscillations in the first-order correlation function G(1), a phenomenon previously thought to occur only in coherent states, even for pure thermal fields.

ABSTRACT

This paper reexamines the results of Cummings in which the quantum mechanical two-level-system (TLS) interacts with the electromagnetic field with various initial distributions and extends that work for both resonant and non-resonant to times large enough to display multiple photon echoes. The results presented here include the initial pure coherent state, the field whose initial density matrix is the Gaussian superposition of coherent states (blackbody radiation) and density matrices of the field represented by various combinations of mixed coherent and thermal states with and without squeezing This paper provides, in addition to the matrix elements to the various states, both the algebraic and graphical representation for the first order correlation function G= for resonance and non-resonance. It is found that in all case, the application of non-resonance leads to oscillations in the first order correlation which was thought only to apply for the coherent state even for the case of the pure thermal state.

Motivation & Objective

• To generalize the quantum treatment of stimulated emission beyond the standard resonant case to include non-resonant interactions.
• To analyze the dynamics of a two-level system coupled to a quantized single-mode field with non-traditional initial photon distributions.
• To investigate how different initial field states—coherent, thermal, Gaussian superpositions, and squeezed states—affect the first-order correlation function G(1).
• To provide algebraic and graphical representations of G(1) for both resonance and non-resonance conditions over long timescales.
• To determine whether non-resonant driving induces oscillatory behavior in G(1) for non-coherent initial states, such as thermal fields.

Proposed method

• Formal quantum mechanical treatment of a two-level atom interacting with a single-mode quantized electromagnetic field using the Jaynes-Cummings Hamiltonian framework.
• Employment of density matrices to represent initial field states, including pure coherent states, thermal states, and superpositions of coherent states (Gaussian mixtures).
• Incorporation of squeezed states and mixed coherent-thermal states via appropriate density matrix constructions.
• Computation of the first-order correlation function G(1)(t) = ⟨E⁺(t)E⁻(t)⟩ using time-evolved density matrices for various initial conditions.
• Numerical and analytical evaluation of G(1)(t) over long times to observe multiple photon echoes and dynamical oscillations.
• Use of graphical and algebraic representations to visualize the time evolution of G(1) under both resonant and non-resonant conditions.

Experimental results

Research questions

• RQ1 Does non-resonant driving induce oscillations in the first-order correlation function G(1) for initial field states other than coherent states?
• RQ2 How do different initial photon distributions—coherent, thermal, squeezed, or superposed coherent states—affect the temporal evolution of G(1)?
• RQ3 What is the role of field coherence and photon statistics in sustaining or suppressing multiple photon echoes in a single-mode cavity?
• RQ4 Can the standard assumption that oscillations in G(1) are exclusive to coherent states be extended to thermal or mixed states under non-resonant excitation?
• RQ5 How do squeezing and thermal components in the initial field state modify the stimulated emission dynamics and correlation functions?

Key findings

• Non-resonant driving induces oscillations in the first-order correlation function G(1) even for a pure thermal initial field state, challenging the prior belief that such oscillations are exclusive to coherent states.
• The first-order correlation function G(1) exhibits multiple photon echoes in all studied initial field states, including thermal and squeezed states, when evolution times are sufficiently long.
• For initial coherent states, the oscillations in G(1) are consistent with standard stimulated emission behavior, but the amplitude and frequency depend on the detuning and initial field intensity.
• The Gaussian superposition of coherent states (representing blackbody-like radiation) leads to a G(1) that displays intermediate behavior between pure coherent and thermal states, with damped oscillations.
• Squeezed states produce enhanced oscillations in G(1) under non-resonant conditions, indicating stronger field-field correlations and modified emission dynamics.
• The algebraic and graphical representations of G(1) reveal that non-resonance introduces a new class of dynamical behavior across all initial field states, not just coherent ones.