Prodiabatic Elimination: Higher Order Elimination of Fast Variables with Quantum Noise

Abstract

We introduce the prodiabatic elimination, a powerful approximation technique that systematically extends the adiabatic elimination of fast degrees of freedom in light-matter coupled systems. Through a controlled expansion of operators, the prodiabatic elimination incorporates higher-order corrections and consistently includes noise contributions, leading to a significantly improved performance compared to standard adiabatic elimination. Importantly, it retains the simplicity and computational efficiency of the adiabatic elimination, making it convenient for practical applications. We demonstrate the approach on two setups: a driven dissipative Jaynes-Cummings model and a three-level system in a two-mode cavity that performs stimulated Raman adiabatic passage (STIRAP). These examples establish the prodiabatic elimination as a robust and broadly applicable tool for analyzing open quantum systems.

Figures (4)

• Figure 1: General setup. A quantum system (purple) is embedded in a driven cavity. The quantum system is assumed to evolve slowly compared to the time-scale governing the cavity, allowing to eliminate the light. The cavity drive is denoted by $f(t)$, its linewidth by $\kappa$, and the quantum system dissipates at rate $\gamma$.
• Figure 2: Prodiabatic elimination for the driven-dissipative Jaynes-Cummings model. (a) Time-dependent behavior of $\langle\hat{\sigma}_z\rangle$ for different drive strengths. Initially the atom is in the ground state. (b) Second order correlation function. Semi-transparent: numerical solution, solid: prodiabatic elimination, dashed adiabatic elimination. The inset in shows the short timescale behavior where the noise contribution is relevant. Parameters: $g/\kappa = 3/20$, $\gamma/\kappa = 5\cdot 10^{-3}$, $\Omega/\kappa=5\cdot 10^{-4}$, $\Delta/\kappa = 1/20$, in (b)$f/\kappa = 5/2\cdot 10^{-4}$.
• Figure 3: Prodiabatic elimination for STIRAP protocol. Semi-transparent: numerical simulations, dashed: adiabatic elimination, solid: prodiabatic elimination and doted: Lindblad master equation Eq. \ref{['eq: LME STIRAP']}. The drives are boxcar functions: $f_{\rm V}(t) = \kappa$ if $|55 -\kappa t|\leq 10$ and $f_{\rm H}(t) = \kappa$ if $|45 -\kappa t|\leq 10$, further $g/\kappa = 1/10$ and $\gamma/\kappa = 5\cdot 10^{-4}$.
• Figure S1: Prodiabatic elimination for the STIRAP setup. Left panels: populations as a function of time. Semi-transparent: numerical simulations, dashed: adiabatic elimination and solid: prodiabatic elimination. The drives are Gaussian $f_i(t) = c_i \exp(-\frac{(t-\tau_i)^2}{2s_i^2})$, where for the top: $c_{\rm H}/\kappa = c_{\rm V}/\kappa = 3/4$, $\kappa s_{\rm H} = \kappa s_{\rm H} = 12$, $\kappa \tau_{\rm V} = 58$ and $\kappa \tau_{\rm H} = 42$. For the bottom: $c_{\rm H}/\kappa = c_{\rm V}/\kappa = 3/4$, $\kappa s_{\rm H} = \kappa s_{\rm H} = 4$, $\kappa \tau_{\rm V} = 52.5$ and $\kappa \tau_{\rm H} = 47.5$. Right panels the corresponding pulses, solid $2f_i/\kappa$ (as they enter the adiabatic elimination) and dashed $F_i$ (as they enter the prodiabatic elimination). System parameters: $g/\kappa = 1/5$ and $\gamma/\kappa = 5\cdot 10^{-4}$