Local energy assignment for two interacting quantum thermal reservoirs

Abstract

Understanding how to assign internal energy, heat, and work in quantum systems beyond weak coupling remains a central problem in quantum thermodynamics, particularly as the difference between competing definitions becomes increasingly relevant. We identify two common sets of definitions for first-law quantities that are used to describe the thermodynamics of quantum systems coupled to thermal environments. Both are conceptually non-symmetric, treating one part of the bipartition (the "system") differently from the other (the "bath"). We analyze these in a setting where such roles are not easily assigned - two large (but finite) sets of thermal harmonic oscillators interacting with each other. We further compare them with a third set of definitions based on a local, conceptually symmetric open-system approach ("minimal dissipation") and discuss their quantitative and structural differences. In particular, we observe that all three sets of definitions differ substantially even when the two subsystems are weakly coupled and far detuned, and that the minimal dissipation approach features distinct work peaks that increase with the coupling strength.

Figures (10)

• Figure 1: Sketch of the model considered in this work: two sets of bosonic modes, all linearly coupled as described by the Hamiltonian of Eq. (\ref{['eq:HNM']}).
• Figure 2: Impact of interaction energy in the dispersive regime, homogeneous frequencies (upper row) and frequency distribution with $\sigma=0.1$ (lower row). (left): bare local energy variations and interaction energy variation, all of the same order of magnitude. (middle): internal-energy variation for subsystem 1 according to the three different definitions. In this case, the bare and minimal dissipation definitions are similar as a consequence of low system temperature. (right): internal-energy variation for subsystem 2 according to the three different definitions. In this case, all definition differ distinctly from each other. Parameters for all figures are $N=200$, $M=300$, $\omega_2 = 0.3 \omega_1$, $g=10^{-5} \omega_1$,$\gamma=10^{-5} \omega_1$, $T_1\omega_1 = 0.6$, $T_2\omega_1 = 4$. Horizontal axis shows time in units of $\omega_1^{-1}$.
• Figure 3: Secondary peaks in the minimal dissipation energy variation, for homogeneous frequencies (upper row) and large distribution of frequencies $\sigma=0.3$ (lower row). (left): internal-energy variation for subsystem 1 according to the three different definitions, at strong coupling $\gamma=2\cdot 10^{-3}\omega_1$. The minimal dissipation definition shows secondary dips which do not appear in the other definitions. (right): internal-energy variation for subsystem 1 according to the three different definitions, at stronger coupling $\gamma=5 \cdot 10^{-3}\omega_1$. The dips in the minimal dissipation definition are drastically more pronounced. Common parameters for the two figures are $N=200$, $M=300$, $\omega_2 = 1.7 \omega_1$, $g=10^{-5} \omega_1$, $T_1\omega_1 = 0.6$, $T_2\omega_1 = 4$. Horizontal axis shows time in units of $\omega_1^{-1}$.
• Figure 4: Impact the detuning sign, homogeneous frequencies. (left column): bare local energy variations and interaction energy variation. (right column): internal-energy variation for subsystem 1 according to the three different definitions. (First row): positive detuning, taking $\omega_2 = 0.3\omega_1$. (Second row): negative detuning, taking $\omega_2 = 1.7\omega_1$. Interaction energy variation has the same sign as the detuning. The minimal dissipation definition is either larger or smaller than the bare definition depending on the detuning sign. Common parameters for all figures are $N=200$, $M=300$, $g=10^{-5} \omega_1$,$\gamma=2\cdot 10^{-3} \omega_1$, $T_1\omega_1 = 0.6$, $T_2\omega_1 = 4$. Horizontal axis shows time in units of $\omega_1^{-1}$.
• Figure 5: Impact of interaction energy in the collective regime, homogeneous frequencies. (left): bare local energy variations and interaction energy variation, all of the same order of magnitude. (middle): internal-energy variation for subsystem 1 according to the three different definitions. (right): internal-energy variation for subsystem 2 according to the three different definitions. Parameters for all three figures are $N=200$, $M=300$, $\omega_2 = 0.3 \omega_1$, $g=10^{-1} \omega_1$,$\gamma=2\cdot 10^{-3} \omega_1$, $T_1\omega_1 = 0.6$, $T_2\omega_1 = 4$. Horizontal axis shows time in units of $\omega_1^{-1}$.