Energetics of self-organization in a dissipative two-site quantum system driven by single-photon pulses

TL;DR

The paper investigates nonequilibrium self-organization in a dissipative two-site quantum system driven by single-photon pulses, extending quantum dissipative adaptation ($QDA$) beyond the $\Lambda$-configuration. It derives a generalized relation in which the absorbed work $W_{\mathrm{abs}}$ (scaled by $\hbar \omega_L$) is a weighted sum of the two $\Lambda$-type transition probabilities, plus a coherence term $\rho_{+-}$ that captures quantum interference between the two excited states. The key finding is that, except in the large-$J$ limit where standard QDA is recovered, the absorbed work does not map linearly to the direct ground-state transition probability; coherence can cause excess work that does not contribute to self-organization, signaling a quantum signature in the energetics. For monochromatic or strongly separated excited states, the standard QDA is recovered and self-organization becomes more robust to finite linewidths, while intermediate coupling $J\sim \Gamma/2$ yields a broadband, coherence-driven deviation. The results also show that with multiple single-photon pulses, the transition probability tends toward a broad plateau, indicating resilience of self-organization to spectral broadening in cascaded photon scenarios.

Abstract

Finding principles of nonequilibrium self-organization in dissipative quantum systems is an open problem. One example is the notion of quantum dissipative adaptation (QDA), that relates the transition probability between the ground states of a quantum system to the nonequilibrium work absorbed during the transition. However, QDA has been originally derived with three-level systems in lambda (Λ) configuration. Here, we consider a model consisting of a two-site system driven by single-photon pulses. We find that the absorbed work is generally related to the sum of Λ-type transition probabilities, instead of the direct transition probability between the two ground states. Although this is equivalent to standard QDA in most scenarios, we find an exception whereby optimal self-organization does not maximize work consumption. We show how quantum coherence leaves this kind of imprint in the energetics of self-organization in the present model.

Figures (12)

• Figure 1: The model. (a) Single-photon pulse added to a zero-temperature environment, as described by the quantum state $|1_a\rangle$, which only interacts with the transition between the states $|g_a\rangle$ and $|e_a\rangle$. (b) Two-site system with ground states $|g_{a,b}\rangle$. Each site has an excited level $|e_{a,b}\rangle$, with energy $\hbar\omega_{a,b}$. The excited levels of the two sites are coherently coupled, at rate $J$. Due to the quantized electromagnetic field, spontaneous emission rates $\Gamma_{a,b}$ appear for the uncoupled sites. (c) At $t=0$, the two ground levels can be equally populated (black circles). The two possible pathways for the system to transition from $|g_a\rangle \rightarrow |g_b\rangle$ are by jumping (quantum coherently) the energy barrier through the eigenstates $|\pm\rangle$, separated by the gap energy $\hbar \omega_{J\delta}$. These form two $\Lambda$-type transitions. The dressed decay rates are denoted by $\Gamma_{a,b}^{(\pm)}$. (d) At $t\rightarrow \infty$, we expect the system to undergo self-organization to the pure state $|g_b\rangle$. Here, we obtain the probability for that transition to happen, as well as the energetics of this process. Our main goal is to test whether the quantum dissipative adaptation relation applies to this model.
• Figure 2: Optimal self-organization driven by maximal work absorption at high intersite couplings, as predicted by the quantum dissipative adaptation relation (Eq.(\ref{['Jinfty']})). Blue full: transition probability $p_{g_a \rightarrow g_b}$ as a function of the detuning $\delta_L^{(-)} = \omega_L - \omega_-$. Black dotted: absorbed work $W_{\mathrm{abs}}/\hbar \omega_L$ as a function of $\delta_L^{(-)}$. We set $\Gamma_b = \Gamma_a$, $\Delta = 0.001 \Gamma_a$, and $2J = \omega_{J\delta} = 10 \Gamma_a$.
• Figure 3: Deviation from standard QDA at the intermediate coupling $J/\Gamma_a = 1/2$: optimal self-organization can be achieved without maximal work consumption. Blue full: transition probability $p_{g_a \rightarrow g_b}$ as a function of $\delta_L^{(-)}$. Black dotted: absorbed work $W_{\mathrm{abs}}/\hbar \omega_L$ as a function of $\delta_L^{(-)}$. We set $\Gamma_b = \Gamma_a$, and $\Delta = 0.001 \Gamma_a$. Inset: broadband-like plateau of the transition probability around $\delta_L^{(-)} = 0.5 \Gamma_a$.
• Figure 4: Robustness to finite linewidth pulses ($\Delta = \Gamma_a$), at $J/\Gamma_a = 0.5$. Blue full: $p_{g_a \rightarrow g_b}$ as a function of $\delta_L^{(-)}$. Black dotted: $W_{\mathrm{abs}}/\hbar \omega_L$ as a function of $\delta_L^{(-)}$. We set $\Gamma_b = \Gamma_a$. Inset: the maximum transition probability achieves around $60\%$.
• Figure 5: Robustness to finite linewidth pulses ($\Delta = \Gamma_a$), at $J/\Gamma_a = 5$. Blue full: $p_{g_a \rightarrow g_b}$ as a function of $\delta_L^{(-)}$. Black dotted: $W_{\mathrm{abs}}/\hbar \omega_L$ as a function of $\delta_L^{(-)}$. We set $\Gamma_b = \Gamma_a$, Inset: the maximum transition probability achieves around $50\%$.