## 5 Quantum Heating and Quantum Cooling Due to Quantum Environment--Kurizki's Model  
A theoretical model addressing explicitly the erasure of system-environment quantum correlations with a specific Hamiltonian mechanism, were proposed by [PERSON] et al. [30] and [PERSON] et al. [31]. These investigations explore the erasure of correlations in the physical context of quantum cooling (and heating) of a two-level system undergoing frequent quantum non-demolition (QND) measurements [32]. Unexpected non-Markovian cooling/heating effects were shown to be related with novel features of coherent dynamics of the interaction between system and bath, which may take place during in times significantly shorter than the memory time of the bath's dynamics. This result is also in line with our present QTD-analysis of the INS experiments, since it emphasizes the importance of the bath being a quantum system participating actively to the system-bath dynamics. Figure 5: Comparison of the two experimental results: Experimental \(S(K,E)\) maps of the two original papers, appropriately stretched/scaled to facilitate comparison. (**a**): data from [29]; (**b**) data from [28]. The shown two vertical dashed lines (white) are guides to the eye. The two broad continuous ribbons due to the translational modes are virtually identically positioned in the momentum-energy plane \(K-E\). The analysis of reference [30] considers the QND measurement of a system \(S\) by a second system \(D\), a detector. [\(D\) corresponds to the neutron, in our INS case]. \(H_{SD}(t)\) is the time-dependent interaction Hamiltonian. Let the total \"system + bath\" Hamiltonian be  
\[H_{tot}=H_{S}+H_{B}+H_{SB} \tag{12}\]  
(\(H_{S}\): system free Hamiltonian; \(H_{B}\): bath Hamiltonian; \(H_{SB}\): system-bath interation). Including also the coupling of \(H_{tot}\) with the detector \(D\), one has:  
\[H(t)=H_{tot}+H_{SD}(t) \tag{13}\]  
The authors consider the evolution during an ultra-short time interval. In this context, it is crucial that the two coupling Hamiltonians \(H_{SB}\) and \(H_{SD}\) are not \"simplified\" to the common rotating-wave approximation of quantum optics [32]. That is, strict energy conservation between the system and the bath or the detector is not imposed [30]. [Recall that also in our INS case, energy conservation does not hold for \"H + neutron\", as repeatedly was pointed out]. Furthermore, the authors showed that if the detector disturbs impulsively the system, immediately after the measurement this action increases the system's energy (which is called \"heating\") [30]. This analytical result was confirmed by accompanying numerical simulations. Let \(\tau\) be the ultra-short time of the (nearly impulsive) quantum measurement (i.e., \(\tau\to 0\)) of the considered two-level system. In more formal terms, and for \(0\leq t\leq\tau\), the authors derived the following results:  
\[\langle H_{SB}(0)\rangle\leq 0 \rightarrow \langle H_{SB}(\tau)\rangle=0 \tag{14}\] \[\langle H_{SD}(t)\rangle = -\langle H_{SB}(t)\rangle\] \[\langle H_{D}\rangle = \text{const} \tag{15}\]  
The energy transfers captured by these relations are associated with changes of the amount of entanglement (and other quantum correlations) between system and bath. This furthermore leads to redistribution of energy and entropy between the different parts of the complete system [30]. Further details of this process are as follows. At sufficiently short times \(t=\tau+\Delta t\) after the measurement, in which holds \(H_{SD}(t\geq\tau)=0\), the common rotating wave approximation does not apply, which implies that \(\langle H_{S}+H_{B}\rangle\) is not conserved, although, of course, \(\langle H_{tot}\rangle\) is conserved. \(\langle H_{SB}\rangle\) decreases towards its undisturbed value.