# What does the master equation for this quantum autonomous absorption refrigerator imply?

I'm following this paper: https://arxiv.org/abs/0908.2076, about building minimal and autonomous quantum absorption refrigerators. The setup is 3 qubits, each coupled to their own baths, and with some interaction at weak coupling allowing for transitions $$|101\rangle \leftrightarrow |010\rangle$$.

In equation (5) of this paper they present a master equation governing the dynamics of the system. The steady-state version of it takes the form $$\dot{\rho_\text{s}} = 0 = -i [H, \rho_\text{s}] + \sum_{i=1}^3 p_i \left[ (\tau_i \otimes \operatorname{tr}_i \rho_\text{s}) - \rho_\text{s} \right] ,$$ where the tensor product is my addition, as part of my understanding of the equation (which can be argued), and $$p_i$$ are coupling variables and $$\tau_i$$ is a thermal state at equilibrium with the baths.

The paper's result is to say that system $$i = 1$$, the target of refrigeration, is driven towards a steady-state at temperature $$T^\text{s}_1 < T_c$$, where $$T_c$$ is the temperature of the bath to which it couples. From my understanding, the process is such that heat currents from the $$i = 1$$ and $$i = 3$$ baths takes the form of quanta exciting the corresponding qubits, the Hamiltonian sending $$|010\rangle \to |101\rangle$$ (biased in this direction due to the choice of temperatures), implying that a quanta of the $$i = 2$$ qubit is absorbed by the corresponding bath, meaning heat goes in that direction. In summary, heat is taken from the cold bath (in conjunction with a "work" bath) and put into the hot bath.

From its construction (which is not very much elaborated in the paper), it seems that each subsystem state is to be substituted by a corresponding thermal state at equilibrium with the bath (the trace part). From this interpretation alone it would mean that qubit 1 is not being cooled but tries to thermalise with the bath, defeating the purpose of the machine. Now, this interpretation isn't complete because there are other parts to the equation.

In light of this, how should I interpret the master equation above? Or what can we take from it? What is the physics that it implies?

This "reset master equation" is a simple phenomenological description of thermalisation. It is based on a (in general, quite unrealistic) model of a system-bath interaction that belongs to the wider family of so-called collision models. The idea is as follows: in each small time interval $$\Delta t$$, there is a small probability $$p_i\Delta t$$ that qubit $$i$$ is swapped with another qubit in a thermal state. You should convince yourself that this brutal reset is described by the completely positive and trace preserving map, $$\hat{\rho} \to \hat{\tau}_i \otimes {\rm tr}_i[\hat{\rho}]$$, where $$\hat{\tau}_i$$ is the thermal state of qubit $$i$$ and $${\rm tr}_i$$ denotes the partial trace. With probability $$1-\sum_ip_i\Delta t$$, no swap occurs and the state instead evolves under the unitary transformation $$\hat{\rho}\to {\rm e}^{{\rm -i}\hat{H}\Delta t}\hat{\rho} {\rm e}^{{\rm i}\hat{H}\Delta t} = \hat{\rho} {\rm -i} \Delta t[\hat{H},\hat{\rho}] + \mathcal{O}(\Delta t^2),$$ in units with $$\hbar=1$$. Therefore, the ensemble-averaged evolution over one small time step is \begin{align}\hat{\rho}(t+\Delta t) & = \left(1 - \sum_i p_i\Delta t \right)\left(\hat{\rho}(t) {\rm -i} \Delta t[\hat{H},\hat{\rho}(t)]\right) + \sum_i p_i \Delta t \, \hat{\tau}_i \otimes {\rm tr}_i[\hat{\rho}(t)] + \mathcal{O}(\Delta t^2) \\ & = \hat{\rho}(t) {\rm -i} \Delta t[\hat{H},\hat{\rho}(t)] + \sum_i p_i \Delta t \,\left( \hat{\tau}_i \otimes {\rm tr}_i[\hat{\rho}(t)] - \hat{\rho}(t)\right) + \mathcal{O}(\Delta t^2), \end{align} which reduces to the quoted master equation in the limit $$\Delta t\to 0$$.
In the absence of any Hamiltonian dynamics, the master equation describes pure exponential relaxation of each qubit to its corresponding thermal state. However, an interaction allows for heat to flow between the qubits. The competition between this interaction and the thermalising baths sets up a non-equilibrium steady state. As pointed out by the OP, when $$p_1\neq 0$$ qubit 1 is in contact with a bath as well as the rest of the machine. If the refrigeration is effective, then the machine's cooling power will surpass the effect of this bath, driving the qubit into a state with local temperature less than $$T_c$$. This is no different to any other refrigerator, which reduces the temperature of a target object or volume below the ambient temperature, thus overcoming the effect of the surroundings that would otherwise warm it back up.
• A couple of questions more, to see if I really understood the details. If $p_i \Delta t$ are probabilities, should they just sum to one in the derivation of the equation (the commutator term for instance is left unchanged)? What's the purpose of this term in the equation's final form? Some sort of term accounting for the possibility of no swap happening?
• $p_i \Delta t$ are probabilities for a swap to occur, but there is also the possibility of no swap (indeed this is the most likely occurence in any small time interval $\Delta t$). So it is certainly not the case that $\sum_i p_i \Delta t=1$. The normalisation condition is $p_{\rm no\, swap} + p_{\rm swap} = (1-\sum_ip_i\Delta t) + \sum_i p_i\Delta t=1$. I have edited to clarify this point. Commented May 18, 2021 at 17:12