Reduction of thermodynamic uncertainty by a virtual qubit

TL;DR

This work analyzes the thermodynamic uncertainty relation (TUR) in a class of quantum thermal machines where a virtual qubit is coherently driven, showing that steady-state currents and entropy production can be captured by a classical Markov process while current fluctuations receive a purely quantum correction from coherence. The total TUR factorizes as $\mathcal{Q}=\mathcal{Q}_d+\mathcal{Q}_c$, with $\mathcal{Q}_d$ obeying the classical bound $\mathcal{Q}_d\ge 2$ and $\mathcal{Q}_c\le 0$ under resonant conditions, enabling $\mathcal{Q}<2$. The authors derive analytical expressions for the minimal uncertainty $\mathcal{Q}^{\min}$ and the optimal coupling strength (characterized by $r$) that maximize steady-state coherence, and they illustrate the results with driven-qubit and two-qubit heat-transport models. Their framework holds for both autonomous and driven machines, and they connect the local and global master-equation descriptions near resonance, highlighting coherence as a quantum resource for enhancing energy-transport precision.

Abstract

The thermodynamic uncertainty relation (TUR) imposes a fundamental constraint between current fluctuations and entropy production, providing a refined formulation of the second law for micro- and nanoscale systems. Quantum violations of the classical TUR reveal genuinely quantum thermodynamic effects, which are essential for improving performance and enabling optimization in quantum technologies. In this work, we analyze the TUR in a class of paradigmatic quantum thermal-machine models whose operation is enabled by coherent coupling between two energy levels forming a virtual qubit. Steady-state coherences are confined to this virtual-qubit subspace, while in the absence of coherent coupling the system satisfies detailed balance with the thermal reservoirs and supports no steady-state heat currents. We show that the steady-state currents and entropy production can be fully reproduced by an effective classical Markov process, whereas current fluctuations acquire an additional purely quantum correction originating from coherence. As a result, the thermodynamic uncertainty naturally decomposes into a classical (diagonal) contribution and a coherent contribution. The latter becomes negative under resonant conditions and reaches its minimum at the coupling strength that maximizes steady-state coherence. We further identify the optimization conditions and the criteria for surpassing the classical TUR bound in the vicinity of the reversible limit.

Figures (7)

• Figure 1: (Color online) Scheme of our device. (a) The thermal machine $M$ is in contact with multiple reservoirs at temperatures $T_1, \ldots, T_N$. Two of its states, $|\Phi_0\rangle$ and $|\Phi_1\rangle$, are coherently coupled through the interaction Hamiltonian $H_I$. When $H_I$ vanishes, the system reaches a detailed-balance steady state, i.e., there is no net transition between any pair of states. The detailed-balance condition requires that, when multiple transition paths exist between two states, the algebraic sum of the numbers of photons with frequency $\omega_i$ emitted to reservoir $i$ (with absorption counted as negative) is identical for all such paths. (b) Example of two transition paths from $|\Phi_k\rangle$ to $|\Phi_l\rangle$. Along both paths, the number of photons with frequency $\omega_1$ emitted to the reservoir characterized by the dissipator $D_1$ is $1$, and the number of photons with frequency $\omega_2$ emitted to the reservoir associated with $D_2$ is $0$.
• Figure 2: (Color online) Schematic illustration of the simplest device converting work into heat. A qubit is weakly coupled to a thermal reservoir at temperature $T$ and driven by a time-dependent external field, with the interaction Hamiltonian given in Eq. (\ref{['HIt']}). In this model, the virtual qubit coincides with the physical qubit itself, i.e., $|\Phi_0\rangle$ and $|\Phi_1\rangle$ correspond to the qubit eigenstates $|0\rangle$ and $|1\rangle$, respectively.
• Figure 3: (Color online) Thermodynamic uncertainty of the driven qubit model. (a) $\mathcal{Q}$ (solid lines) and its diagonal contribution $\mathcal{Q}_d$ (dashed lines) as functions of $r/r_0$ at $\Delta=0$. Red, blue, and green curves correspond to $r_0 = 0.947$, $0.5$, and $0.97$, respectively. (b) $\mathcal{Q}$ (solid lines) and $\mathcal{Q}_d$ (dashed lines) as functions of $r_0$ at $\Delta=0$, with $r = 0.5\, r_0$ (red) and $r = \frac{3-\sqrt{5}}{6}\, r_0$ (blue).
• Figure 4: (Color online) Schematic illustration of heat transport mediated by two interacting qubits. Each qubit is weakly coupled to its own thermal reservoir at temperatures $T_1$ and $T_2$, respectively. The two qubits are coupled through a time-independent interaction of the form given in Eq. (\ref{['HIt']}), corresponding to $\omega_d = 0$. In this model, the virtual qubit is formed by the two degenerate energy levels of the composite system, namely $|\Phi_0\rangle = |01\rangle$ and $|\Phi_1\rangle = |10\rangle$.
• Figure 5: (Color online) Thermodynamic uncertainty in the two-qubit heat-transport model with $p_1=p_2$. (a) The total thermodynamic uncertainty $\mathcal{Q}$ (solid lines) and its diagonal contribution $\mathcal{Q}_d$ (dashed lines) as functions of $r/r_0$. The red curve corresponds to $R_{1,2}=(1\pm r_0)/2$ with $r_0=0.835$, while the blue curve corresponds to $R_1=1/2$ and $R_2=1/2-r_0$ with $r_0=0.259$. (b) $\mathcal{Q}$ (solid lines) and $\mathcal{Q}_d$ (dashed lines) as functions of $r_0$, where $r=3r_0/8$. The red and blue curves correspond to $R_{1,2}=(1\pm r_0)/2$ and $R_1=1/2$, $R_2=1/2-r_0$, respectively.