Peltier Cooling of Fermionic Quantum Gases

We propose a cooling scheme for fermionic quantum gases, based on the principles of the Peltier thermoelectric effect and energy filtering. The system to be cooled is connected to another harmonically trapped gas acting as a reservoir. The cooling is achieved by two simultaneous processes: (i) the system is evaporatively cooled, and (ii) cold fermions from deep below the Fermi surface of the reservoir are injected below the Fermi level of the system, in order to fill the “holes” in the energy distribution. This is achieved by a suitable energy dependence of the transmission coefficient connecting the system to the reservoir. The two processes can be viewed as simultaneous evaporative cooling of particles and holes. We show that both a significantly lower entropy per particle and faster cooling rate can be achieved in this way than by using only evaporative cooling.

Figure 1

(a) Sketch of the proposed Peltier cooling scheme: atoms are injected from deep energy levels of the reservoir cloud ($R$) into the system cloud ($S$) just below the Fermi level through a channel with an energy-dependent transmission $\mathcal{T}(ϵ)$. Additionally, the system is submitted to evaporative cooling with a fixed evaporation threshold ${ϵ}_1$ above Fermi level, removing hot particles. (b) Evolution of the Fermi distribution of the system at three stages during the cooling process: initial (dashed blue curve, $T_S\approx T_{FS}$), intermediate (dotted blue curve, $T_S=0.3T_{FS}$), and final (solid blue curve, $T_S=0.02T_{FS}$). The evolution, indicated by arrows, is calculated (see text) for ${ϵ}_1=1.05E_{FS}^0$, ${\gamma }_{\mathrm{ev}}{\tau }_0=15$, ${ϵ}_0=0.99E_{FS}^0$, $\mathrm{\Delta }ϵ=0.96E_{FS}^0$, and $E_{FS}^0=0.25E_{FR}^0$. The blue and grey shaded regions indicate the injection and evaporation energy windows, respectively. The dashed-dotted curve is the final distribution of the reservoir. (c) Sketch of a Peltier cooling module. The $n$-like and $p$-like thermoelectric materials ensure the transport of low energy (electrons) and high energy (holes) particles, which, thus, carry heat from the cold (blue) to the hot (red) region.

Figure 2

Evolution during the cooling process of (a) the entropy per particle $T_S(t)/T_{FS}(t)$ and (b) the Fermi energy $E_{FS}(t)$ (left axis) and particle number $N_S(t)$ (right axis). The solid curves are for the Peltier cooling with $E_{FS}^0/E_{FR}^0=1/4$ and ${ϵ}_1=1.05E_{FS}^0$, ${\gamma }_{\mathrm{ev}}{\tau }_0=15$, ${ϵ}_0=0.99E_{FS}^0$, $\mathrm{\Delta }ϵ=0.96E_{FS}^0$. The dashed-dotted curves are for evaporative cooling only, with an initial particle number $N=N_S+N_R$ and ${ϵ}_1=1.05E_F$. Inset: minimum entropy per particle achieved by Peltier cooling, as a function of ${ϵ}_0$, for: ${ϵ}_1=1.05E_{FS}^0$ and $({\epsilon }_0-\mathrm{\Delta }ϵ)/E_{FS}^0=3\%$ (red triangles), 20% (purple circles), and ${ϵ}_1=1.5E_{FS}^0$, $({\epsilon }_0-\mathrm{\Delta }ϵ)/E_{FS}^0=3\%$ (blue squares).

Figure 3

Dimensionless cooling rate $\eta (t){\tau }_0$ as a function of $T_S/T_{FS}$, for the same parameters as in Fig. 2. The black dashed curve is for evaporative cooling only. Arrows indicate the direction of the time evolution. The horizontal (red) line indicates a typical heating rate (see, e.g., [43]) limiting these cooling processes.

Figure 4

(a) Dimensionless cooling rate $\eta (t){\tau }_0$ as a function of $T_S/T_{FS}$, for $\mathrm{\Delta }ϵ=0.96E_{FS}^0$ and various transmissions centered at ${ϵ}_0=0.99E_{FS}^0$ (the other parameters are taken as in Fig. 2): The (red) dotted-dashed and (orange) dashed curve correspond to a single and 100 parallel resonant level(s), respectively, with $\mathrm{\Gamma }=1·{10}^{-3}{E}_{FS}^0$. The (blue) dotted curve is for two resonant levels in series of width $\mathrm{\Gamma }=0.03E_{FS}^0$ and the (green) solid curve is for an ideal box transmission. The (black) dashed-dotted curve shows the evaporative cooling only. (b) The corresponding energy-dependent transmission coefficients. The grey area indicates states above ${ϵ}_1$, while the blue one indicates those below $\mathrm{\Delta }ϵ$, which do not participate in transport.