[Paper Review] Dispersive Jaynes-Cummings Hamiltonian describing a two-level atom interacting with a two-level single mode field

C. J. S. Ferreira, Clodoaldo Valverde+1 more|arXiv (Cornell University)|Sep 22, 2016

Quantum Information and Cryptography40 references3 citations

TL;DR

This paper investigates the time evolution of a two-level atom interacting with a two-mode single-field via a dispersive Jaynes-Cummings Hamiltonian, showing that the field exhibits periodic energy exchange between two Fock states |n₁⟩ and |n₂⟩ with a period dependent on the ratio n₂/n₁. The statistical properties of the field transition between sub-Poissonian and super-Poissonian regimes depending on n₁ and n₂, with no energy exchange but phase entanglement between atom and field.

ABSTRACT

We investigate the time evolution of statistical properties of a single mode radiation field after its interaction with a two-level atom. The entire system is described by a dispersive Jaynes-Cummings Hamiltonian assuming the atomic state evolving from an initial superposition of its excited and ground states, $\vert e angle +\vert g angle ,$ and the field evolving from an initial superposition of two excited levels, $\vert n_{1} angle+ \vert n_{2} angle$. It is found that the field evolution is periodic, the period depending on the ratio $n_{2}/n_{1}.$ The energy excitation oscillates between these two states and the statististics can be either sub- or super-Poissonian, depending on the values $n_{1},$ $n_{2}$.

Motivation & Objective

• To analyze the time evolution of statistical properties in a quantized single-mode field interacting with a two-level atom via a dispersive Jaynes-Cummings Hamiltonian.
• To investigate how initial superpositions of two Fock states |n₁⟩ and |n₂⟩ evolve under dispersive interaction.
• To determine the conditions under which the field statistics become sub-Poissonian or super-Poissonian.
• To explore the feasibility of preparing initial superposed field states such as (|n₁⟩ + |n₂⟩)/√2 using atomic cascades in a cavity.
• To examine the absence of field squeezing when only two Fock components are involved.

Proposed method

• Formulates the dispersive Jaynes-Cummings Hamiltonian with detuning Δω ≫ λ, leading to an effective interaction Hamiltonian V′′ = λ′n(|i⟩⟨i| + |e⟩⟨e|), where λ′ depends on λ and Δω.
• Assumes initial atomic state |ψ_A⟩ = (|e⟩ + |g⟩)/√2 and initial field state |ψ_F⟩ = (|n₁⟩ + |n₂⟩)/√2.
• Derives the time-evolved atom-field state using the unitary evolution operator Û(t) = exp(−itV′′), resulting in an entangled state |ψ_AF(t)⟩ = |g⟩|ψ_F⟩ + |e⟩e^{iϕ(t)ⁿ}|ψ_F⟩.
• Analyzes the photon number distribution Pₙ(t) and Mandel parameter to assess statistical behavior (sub-Poissonian or super-Poissonian).
• Uses analytical expressions for the field state evolution and probability amplitudes, including the phase factor ϕ(t) = λ′t.
• Extends results to non-equal weight superpositions and discusses experimental preparation via N-atom cascades in a cavity, referencing prior work on generating Schrödinger-cat-like states.

Experimental results

Research questions

• RQ1 How does the field evolve statistically when interacting with a two-level atom in a dispersive regime with initial superposition of two Fock states?
• RQ2 What determines the periodicity of energy oscillation between |n₁⟩ and |n₂⟩ in the field state?
• RQ3 Under what conditions does the field exhibit sub-Poissonian or super-Poissonian statistics?
• RQ4 Can initial superpositions of two Fock states be experimentally prepared using atomic cascades in a cavity?
• RQ5 Why is no field squeezing observed when only two Fock components are involved in the superposition?

Key findings

• The field state evolves periodically with a period dependent on the ratio n₂/n₁, with oscillations of energy between |n₁⟩ and |n₂⟩.
• The Mandel parameter indicates sub-Poissonian statistics for certain n₁ and n₂ values, as shown in Fig. 2(a), while super-Poissonian statistics are observed in Figs. 2(b) and 2(c).
• The degree of super-Poissonian character increases with larger differences between n₁ and n₂ (n₂ − n₁ ≫ 1).
• The statistical behavior for pairs (n₁, n₂) and (n₁′, n₂′) with n₁′/n₁ = n₂′/n₂ = p shows similar dynamics but with scaled periods τ′ = τ/p.
• No field squeezing is observed because the state distribution is limited to only two Fock components, consistent with Mandel’s criterion.
• Initial superposed states like (|n₁⟩ + |n₂⟩)/√2 can be generated via N-atom cascades in a cavity, as demonstrated by analytical expressions and numerical results in Fig. 3.