Sub-Bath Cooling in Bosonic Systems: Gaussian Constraints and Non-Gaussian Enhancements

TL;DR

The paper addresses cooling of continuous-variable bosonic systems under finite resources by framing a CV HBAC framework that separates Gaussian and non-Gaussian resources. It derives a tight bound and an optimal Gaussian recharging strategy (successive swaps with increasing machine gaps) showing Gaussian cooling cannot surpass the bath limit and cannot benefit from memory effects, with entropy production scaling as $1/N$ for large machines. It then demonstrates that non-Gaussian $p$-excitation exchange interactions provide a genuine cooling enhancement, yielding a sub-bath cooling condition $p ext{ω}_1> ext{ω}_0$ in single-shot and, in iterative use, a fixed-point cooling limit $eta^*=rac{ ext{ω}_1}{ ext{ω}_0}peta$, with asymptotic mean excitation $ar{n}_S^{( ext{∞})}=rac{1}{e^{peta ext{ω}_1}-1}$. The results establish fundamental CV cooling limits, reveal the pivotal role of non-Gaussianity, and quantify how nonlinearity $p$ accelerates cooling and relaxes machine-frequency requirements, informing experimental design for CV quantum technologies.

Abstract

Cooling quantum systems with finite resources is a central task in quantum technologies and has been extensively explored in discrete-variable settings. As continuous-variable (CV) platforms play an increasingly important role in quantum information processing, it becomes crucial to understand the fundamental limitations of cooling bosonic systems. In this work, we develop a general framework for cooling CV systems, identifying both the constraints imposed by Gaussianity and the advantages enabled by non-Gaussian interactions. We derive a reachable bound on the cooling performance of Gaussian operations that applies to arbitrary cooling architectures. By optimizing over all protocols saturating this bound, we further identify the most efficient scheme, which minimizes dissipated energy for a given number of ancilla modes. Beyond Gaussian operations, we show that $p$-excitation exchange exploits non-Gaussian resources to achieve a $p$-fold enhancement of the cooling limit. Our results establish the fundamental limits of CV heat-bath algorithmic cooling and reveal the crucial role of non-Gaussianity in surpassing Gaussian cooling barriers.

Figures (6)

• Figure 1: A continuous-variable HBAC framework. Here, $V$ denotes the recharging unitary; $U$ and $\Gamma$ denote the thermalizing unitary and the full thermalization channel, respectively. (a) The first round of the most general Gaussian cooling scheme using $N$ ancillas; inside $V$, we represent our result that the most efficient recharging routine is given by successive full $\textsc{swap}$, the single-excitation exchange described in (c). (b) The example of iterative cooling with non-Gaussian operations studied in this paper: the recharging routine is a $p$-excitation exchange operation for $p\geq 2$ illustrated in (d)
• Figure 2: Plot of $\Sigma_N^{**}$ versus $\lambda$, with the initial mean excitation number fixed at $n_0 = 10$. For each $\lambda$, $\Sigma_N^{**}$ is optimized over energy gap of $N = {1, 2, 4}$ machine modes.
• Figure 3: Plot of ${\bar{n}_S(t), \bar{n}_M(t)}$ with initial condition (from left to right) $\bar{n}_S < \bar{n}_{M}$, $\bar{n}_S = \bar{n}_{M}$, $\bar{n}_S > \bar{n}_{M}$. The orange (teal) solid line indicates $\bar{n}_S(t)$ ($\bar{n}_{M}(t)$). The dashed line indicates the initial mean excitation number where $\bar{n}_S=2$, $\bar{n}_M={1.5,2.0,2.5}$.
• Figure 4: The time evolution of $\bar{n}_S^{(L)}:=\bar{n}(L t )$ under iterative cooling for $L=1,2,3,...,2\times 10^4$ and $t=5\times10^{-3} s$. Within each iteration, the collisional channel is governed by $p$-excitation exchange operation. Initially, $\bar{n}_S^{(0)} = 2$, and $\bar{n}_M^{(0)}=1.5$. In the large $L$ limit, the minimum achievable mean excitation saturates to Eq. (\ref{['eq:emanPhotonLB']}), indicated by the dashed line. As $p$-nonlinearity increases, the minimum mean excitation decreases. For $p\geq 2$, the asymptotic mean excitation decreases blow $\bar{n}_M^{(0)}$.
• Figure 5: (a) Trajectory $g$ vs $t$ for cooling from $\bar{n}_s(0) =10$ to $\bar{n}_s(1) =10^{-5}$, which corresponds to $\lambda = 120.8$. The red markers represents optimized energy gap of the machine for 5 timesteps. The blue markers are the discrete solution sampled from the analytical result. (b) Entropy production vs $N$, where $\Sigma_N^{**} (\Sigma_N^{*})$ is the entropy production with numerical solution (discrete timestep solution sampled from the analytical curves)

Theorems & Definitions (19)

• Definition 1
• Lemma 1
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 2