Impossibility of Refrigeration and Engine Operation in Minimal Qubit Repeated-Interaction Models

Abstract

We investigate the operation of a qubit as a quantum thermal device within the repeated interaction framework, allowing for strong system-bath coupling and finite interaction times. We analyze two minimal models: an alternating-coupling setup, in which the qubit sequentially interacts with hot and cold baths, and a simultaneous-coupling setup, where both baths interact with the qubit during each collision. For the alternating model, we obtain an exact analytical solution for the limit-cycle state, valid for arbitrary coupling strengths and collision durations. Using this solution, we rigorously prove a no-go theorem for quantum refrigeration. We further demonstrate that, although work can be generated locally at individual system-bath contacts, the total work over a cycle is always nonpositive, precluding engine operation. In the absence of work, the model describes pure heat conduction, for which we derive a closed-form expression for the heat current and show that it exhibits a nonmonotonic turnover behavior. The simultaneous-coupling model is analyzed perturbatively. In the short-collision-time limit, it reproduces the same steady-state behavior as the alternating model, reinforcing the generality of the constraints identified. Our results establish fundamental limitations on qubit-based quantum thermal machines operating under Markovian repeated interactions and highlight the need for enriched models to realize functional quantum thermal devices.

Figures (17)

• Figure 1: Alternating model for a quantum thermal machine. A two level system ($\hat{H}_S$) collides with each bath separately, alternating between collisions with (a) hot and (b) cold ancillas.
• Figure 2: System relaxation dynamics in the alternating-coupling model from a certain initial state toward the limit cycle with $n$ as the number of collisions. (a) Relaxation of the system's ground-state population to limit cycle values, alternating between $p_S^{(\infty,C)}$ and $p_S^{(\infty,H)}$. As a reference, the populations of ancillas from the baths are shown as dash-dotted lines, blue for the cold bath, $p_C$, and red for the hot bath, $p_H$. (b) Decoherence dynamics in the qubit system. We consider asymmetric coupling strengths to the baths, $J^{H}_{xx} =4$, $J^{H}_{yy} = 16$ and $J^{C}_{xx} = 2$, $J^{C}_{yy} = 8$. The system and ancilla's frequencies are set to $\omega_{S} = \omega_{C} = \omega_{H} = 1$, and the cold (hot) bath inverse temperatures are $\beta_{C} = 2$ ($\beta_{H} = 1$).
• Figure 3: Limit cycle of the alternating-coupling model: Ground-state population of the qubit system as a function of the collision time $\tau$. (a) Symmetric coupling, with $J_{xx}^{C} = J_{yy}^{C} = 5$ and $J_{xx}^{H} = J_{yy}^{H} = 5$. (b) Asymmetric coupling, with $J_{xx}^{C} = J_{yy}^{C} = 5$ and $J_{xx}^{H} = J_{yy}^{H} = 10$. (c) General coupling, with $J_{xx}^{C} =1$, $J_{yy}^{C} = 2$, $J_{xx}^{H}=4$ , and $J_{yy}^{H} = 3$. Other parameters are $\omega_{S} = \omega_{C} = \omega_{H} = 1$, $\beta_{C} = 2$ and $\beta_{H} = 1$. For reference, the ground-state populations of the cold and the hot bath ancillas are indicated by blue (cold, $p_C$) and red (hot, $p_H$) dotted lines.
• Figure 4: Heat exchange at the limit cycle in the alternating-coupling model. Shown is the heat exchanged in a single collision between the system and the cold ancilla at the limit cycle, as given by Eq. (\ref{['eq:QA']}). We consider $J^{A}_{xx} = J^{A}_{yy} \equiv J$ with $A = H, C$, and test cases with asymmetrical couplings at the two ends. Other parameters are $\omega_{A} = \omega_{S} = 1$, $\tau = 0.5$, $\beta_{C} = 2$, and $\beta_{H} = 1$.
• Figure 5: Alternating-coupling model at limit cycle: (a) heat dissipated at the cold ancilla within each collision of duration $\tau$, (b) heat current obtained as the $\tau$ derivative of panel (a). We set $\tau=0.05$, $\omega_S=\omega_{A} = 1$, $\beta_C = 2$, and $\beta_H = 1$.