Heat due to system-reservoir correlations in thermal equilibrium

The heat flow between a quantum system and its reservoir is analyzed when initially both are in a separable thermal state and asymptotically approach a correlated equilibrium. General findings are illustrated for specific systems and various classes of non-Markovian reservoirs relevant for solid state realizations. System-bath correlations are shown to be substantial at low temperatures even in the weak coupling regime. As a consequence, predictions of work and heat for actual experiments obtained within conventional perturbative approaches may often be questionable. Correlations induce characteristic imprints in heat capacities which opens a proposal to measure them in solid state devices.

Figure 1

Heat $Q_{\mathrm{corr}}$ from (15) exchanged between system and an Ohmic-type reservoir to establish equilibrium correlations from an initially factorized state vs inverse temperature $\theta =\Delta \hslash \beta$ for various cutoff frequencies ${\omega }_c/\Delta =$ 20 (blue, dotted), 50 (purple, short-dashed), 100 (green, long-dashed), 200 (red, solid), and coupling constant $\eta =0.1$. Heat is scaled with $\hslash \Delta$.

Figure 2

Heat $Q_{\mathrm{corr}}$ exchanged between system and sub-Ohmic reservoirs with spectral exponents $s$ [see (19)] to establish equilibrium correlations from an initially factorized state at various inverse temperatures $\Delta \hslash \beta =$ 0.1 (red, dotted), 0.5 (green, short-dashed), 1 (purple, long-dashed), and 10 (blue, solid). Heat is scaled with $\hslash \Delta$ and $\eta =0.1,{\omega }_c/\Delta =50$.

Figure 3

Approximate value of the exchanged heat for weak coupling $Q_{\mathrm{corr},\mathrm{app}}$ for a harmonic oscillator normalized to the exact value $Q_{\mathrm{corr},\mathrm{app}}$ vs inverse temperature $\theta$ in a Drude bath $\left({\omega }_c/{\omega }_0=50\right)$. Depicted are results for various coupling strengths $\eta =0.01$ (black, solid), 0.05 (blue, long-dashed), 0.2 (purple, short-dashed), 0.5 (green, dotted–closely spaced), and 1 (red, dotted).

Figure 4

Ratio of the heat capacities $C_{\beta }/C_S$ of correlated and bare thermal state vs inverse temperature for a two-level system and various bath cutoff frequencies ${\omega }_c/\Delta =$ 20 (blue, short-dashed), 50 (green, long-dashed), and 100 (red, solid). For comparison also results for a harmonic system are depicted (purple, dotted) for ${\omega }_c/{\omega }_0=50$. Coupling strength is $\eta =0.1$.

Figure 5

Ratio $C_{\beta }/C_S$ of correlated and bare thermal state of a harmonic oscillator vs coupling strength for various inverse temperatures ${\omega }_0\hslash \beta =$ 0.1 (red, dotted), 1 (green, short-dashed), 4 (purple, long-dashed), and 6 (blue, solid).