Thermal production, protection, and heat exchange of quantum coherences

We consider finite-sized atomic systems with varying number of particles which have dipolar interactions among them and are also under the collective driving and dissipative effect of a thermal photon environment. Focusing on the simple case of two atoms, we investigate the impact of different parameters of the model on the coherence contained in the system. We observe that, even though the system is initialized in a completely incoherent state, it evolves to a state with a finite amount of coherence and preserves that coherence in the long-time limit in the presence of thermal photons. We propose a scheme to utilize the created coherence in order to change the thermal state of a single two-level atom by having it repeatedly interact with a coherent atomic beam. Finally, we discuss the scaling of coherence as a function of the number of particles in our system up to $N=7$.

Figure 1

Schematic view of (a) creation and (b) harvesting of coherences in our model. A pair of two-level atoms initially in a state with coherence $C_L$ can be transformed into another state with higher coherence $C_H>C_L$ by collective interaction with a thermal bath. The energetic cost of the generation of coherences is paid by the “natural” heat $\mathrm{\Delta }Q$ withdrawn from the thermal bath. Some amount of this energy can be harvested back by a repeated interaction method where a single two-level atom is coupled sequentially with a pair of atoms with coherence $C_H$. The coupling happens at random times and the atom reaches a steady state eventually that can be described by a thermal state with an effective temperature $T_{\text{eff}}$ that can be controlled by the $C_H$ of the subenvironment atomic pairs. (a) Heat is used to produce (“charge”) coherences. (b) Coherences are harvested back (“discharged”) as heat.

Figure 2

$C_{l_1}$ as a function of scaled time ${\gamma }_{0}t$ for different mean number of photons, $\overline{n}$, in the environment for the two atoms initially in the ground state.

Figure 3

Dynamics of $C_{l_1}$ for the initial state (a) $|\psi \left(0\right)\rangle =\left(\right|ge\rangle +i|eg\rangle )/\sqrt{2}$ and (b) $|\psi \left(0\right)\rangle =\left(\sqrt{3}\right|ge\rangle +|eg\rangle )/2$. Solid and dashed lines represent $f_0/{\gamma }_0={10}^2$ and $f_0/{\gamma }_0=1$ cases, respectively, with $\overline{n}=10$. Time is dimensionless and scaled with ${\gamma }_0$.

Figure 4

Scaling of $l_1$ norm of coherence in the long-time limit with the number of atoms with an initial state where all the atoms are in their initial state. The dots represent the actual value of the coherence measure while the solid line is the cubic fit to these points. Model parameters are set to $f_0/{\gamma }_0=1$ and $n=10$.