[Paper Review] AMENDART in Markovian circuit QED

А. В. Додонов|arXiv (Cornell University)|Aug 3, 2010

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper investigates the asymptotic mean excitation numbers of cavity fields and atoms in circuit QED systems under Markovian damping, showing that anti-rotating terms (AMENDART) lead to non-zero excitation levels even at zero temperature. Using a generalized master equation that includes anti-rotating cavity-reservoir interactions, it demonstrates that the vacuum state is never exactly achieved, resulting in intrinsic photon and atomic excitation numbers above zero due to dissipative processes, with contributions from parasitic elements like off-resonant atoms and higher cavity modes.

ABSTRACT

We study the cavity field's and atomic asymptotic mean excitation numbers due to anti-rotating term (AMENDART) in the circuit Quantum Electrodynamics (circuit QED) system, composed of a two-level atom and a single cavity field mode, subject to Markovian damping and dephasing mechanisms. We show that the AMENDART are above the thermal values, and their behavior is analyzed analytically and numerically for typical parameters in circuit QED implementations described by the Rabi Hamiltonian. We point out that "parasitic elements", such as other cavity modes or eventual off-resonant atoms, also contribute substantially to AMENDART.

Motivation & Objective

• To analyze the asymptotic mean excitation numbers of cavity fields and two-level atoms in circuit QED under Markovian damping and dephasing.
• To investigate the role of anti-rotating terms (AMENDART) in sustaining non-zero excitation levels in the presence of dissipation.
• To assess the contribution of parasitic elements—such as other cavity modes and off-resonant atoms—to AMENDART in realistic circuit QED implementations.
• To demonstrate that the standard master equation (SME) fails to achieve the true vacuum state due to neglected anti-rotating reservoir interactions.
• To establish that the vacuum state is unattainable under Markovian approximation when anti-rotating terms in system-reservoir coupling are included.

Proposed method

• Formulates a generalized master equation with a Lindblad-type dissipator that includes anti-rotating terms in the cavity-reservoir interaction, preserving positivity and normalization.
• Uses the Rabi Hamiltonian as the effective system Hamiltonian, including both rotating and anti-rotating terms, and applies the Markovian approximation to derive the dynamics.
• Derives analytical expressions for the asymptotic mean photon number ⟨n₁⟩ and atomic excitation number ⟨E₁⟩ using a bilinear form of quadrature operators x and p.
• Applies the condition DₚDₓ − D_z² ≥ (κ/4)² to ensure the physical validity of the dissipative superoperator and its reducibility to Lindblad form.
• Compares the generalized master equation with the standard master equation (SME), showing that the SME neglects anti-rotating reservoir interactions and thus fails to reach the true vacuum.
• Performs analytical and numerical analysis of the asymptotic excitation numbers for typical circuit QED parameters, including g, Ω, κ, λ, and γ.

Experimental results

Research questions

• RQ1 Does the inclusion of anti-rotating terms in the cavity-reservoir interaction lead to non-zero asymptotic excitation numbers in circuit QED under Markovian dynamics?
• RQ2 To what extent do parasitic elements such as higher cavity modes and off-resonant atoms contribute to AMENDART in realistic circuit QED systems?
• RQ3 Can the standard master equation (SME) accurately describe the vacuum state in circuit QED when anti-rotating reservoir interactions are neglected?
• RQ4 What is the analytical form of the asymptotic mean photon and atomic excitation numbers when anti-rotating terms are included in the dissipative dynamics?
• RQ5 Is the vacuum state physically achievable in a Markovian circuit QED system when anti-rotating terms in the system-reservoir coupling are considered?

Key findings

• The asymptotic mean photon number ⟨n₁⟩ and atomic excitation number ⟨E₁⟩ are strictly greater than zero, even at zero temperature, due to anti-rotating terms in the system-reservoir interaction.
• The vacuum state is never exactly achieved in the Markovian approximation when anti-rotating cavity-reservoir interactions are included, as shown by the condition μ = 0 being required for vacuum stability.
• Parasitic elements such as other cavity modes and off-resonant atoms contribute substantially to AMENDART, with typical values on the order of 10⁻³ for coupling strengths within a few percent of the cavity frequency.
• The standard master equation (SME) is insufficient to describe the true vacuum, as it neglects anti-rotating reservoir terms and thus fails to account for intrinsic residual excitations.
• The generalized master equation with bilinear dissipators preserves positivity and normalization, and its solution confirms that non-zero excitation levels persist asymptotically.
• Analytical expressions for ⟨n₁⟩ and ⟨E₁⟩ are derived and shown to be positive under all physically valid parameter choices, confirming the inevitability of residual excitations.