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Comptes Rendus Physique
Volume 17, n° 10
pages 1161-1174 (décembre 2016)
Doi : 10.1016/j.crhy.2016.08.013
Thermoelectric transport and Peltier cooling of cold atomic gases
Transport themoélectrique et refroidissement Peltier dans des gaz d'atomes froids
 

Fig. 1




Fig. 1 : 

Setup for thermoelectricity (sketch). The two reservoirs (h for hot, c for cold) exchange particles and heat through a junction characterized by its linear response properties represented by the matrix   defined in the main text. The thermodynamic response of each reservoir is represented by the coefficients κ , γ and C N , which denote respectively the compressibility, the dilatation coefficient and the heat capacity at constant particle number (see text).


Fig. 2




Fig. 2 : 

Nanostructuration with cold atoms. The colors indicate the total thermopower α as a function of the transverse confinement frequencies ν x and ν z in the channel, for a fixed Fermi temperature T F =μK . a) At low temperature T /T F =0.05, oscillations are visible. Gray lines indicate vanishing total thermopower. b) At high temperature T /T F =0.3, the oscillations are washed out.


Fig. 3




Fig. 3 : 

Temperature and population imbalances, adapted from [27]. a) Temperature evolution: T h (red) and T c (blue) as a function of time for ν z =3.5 kHz and a disorder of average strength 542 nK (see main text). Dashed line:   at the initial time. The average temperature does not evolve significantly with time, in agreement with linear response. b) Population relative imbalance ΔN /N tot as a function of time for a disordered channel in which ν z was set to 3.5 kHz with a disorder of average strength 542 nK (see main text). c) and d) Time evolution of ΔN /N tot a ballistic channel with ν z =3.5 kHz (c)) and ν z =9.3 kHz (d)).


Fig. 4




Fig. 4 : 

Ballistic-diffusive crossover, adapted from [27]. a): Time evolution of ΔN /N tot for a fixed confinement of ν z =3.5 kHz across the ballistic-diffusive crossover, with increasing disorder strength (  and 1.08 μK from bottom to top). Solid lines: theory obtained with the transparency in Eq. (19). b): Fitted timescale τ 0 (black circles) and   (red triangles) as a function of disorder strength for the data set in a). c): Maximal response   versus timescale τ 0 for the diffusive (gray squares) and ballistic (open circles) cases. d): Thermoelectric response   in the regime of strong disorder from   (gray circles) to 1220 nK (black diamonds) and fixed ν z =4.95 kHz, in which the time dependence has been rescaled by τ 0 . Black line: theoretical calculations.


Fig. 5




Fig. 5 : 

Heat-to-work conversion, adapted from [27]. a) Thermodynamic cycle performed by the system in the μN plane for ν z =3.5 kHz and  . The evolution of the hot reservoir is depicted in red and that of the cold one in blue. The solid lines are given by the theory, and the black arrows indicate the direction of time. The two gray triangles indicate the turning points at which the conversion process ends. b), d) Efficiency, power of the channel in the ballistic case, as a function of confinement. c), e) The same quantities as functions of disorder strength for ν z =3.5 kHz. Orange symbols: experiments; grey area: theory. f), g) Dimensionless figure of merit ZT as a function of confinement and disorder for ballistic and diffusive channels, respectively.


Fig. 6




Fig. 6 : 

Principle of Peltier cooling for cold atoms. a) Adapted from [48]. 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  . Additionally, the system is submitted to evaporative cooling with a fixed evaporation threshold ε 1 located above the Fermi level, removing high-energy particles, as indicated in the grey area above S . b) Evolution of the Fermi distribution of the system at three stages during the cooling process: initial (dashed red curve, T S T FS ), intermediate (purple dotted curve, T S =0.3T FS ) and final (solid blue curve, T S =0.02T FS ). The evolution, indicated by arrows, is calculated for  , γ ev τ 0 =1/16,  ,   and  , for an ideal sharp transmission (see below). The blue and grey shaded regions indicate the injection and evaporation energy windows, respectively.


Fig. 7




Fig. 7 : 

a) Sketches of the different propositions to realize the desired energy-dependent transmission functions. Top (red dot): a single resonant level, with a transmission given by (24). Middle (green dots): two resonant levels in series connected by a tunneling probability, generating a transmission given by (25). Bottom (orange dots): several resonant levels in parallel. b) Adapted from [48]. The corresponding energy-dependent transmission coefficients, plotted for parameters identical to those in Fig. 8. The light grey area indicates states above ε 1 , which are subject to evaporation, while the dark grey one indicates those below Δε , which do not participate in transport. The light blue area corresponds to the ideal box-like transmission, bounded by the band bottom and ε 0 . The transmission for three resonant levels in series is shown in purple for comparison with that of the two levels in series. c) Sketch of the energy levels for two resonant levels in series.


Fig. 8




Fig. 8 : 

Adapted from [48]. a) Dimensionless cooling rate η (t )τ 0 as a function of T S /T FS , for   and various transmissions centered at  , and  ,  , γ ev τ 0 =15,  : The (red) dot-dashed and (orange) dashed curves correspond to a single and 100 parallel resonant level(s), respectively, with  . The (green) dotted curve is for two resonant levels in series of width   and the (light blue) solid curve is for an ideal box transmission. (Black) dashed-dotted curve shows the evaporative cooling only. b) Dimensionless cooling rate as a function of the entropy per particle, for the same parameter values as in a). The arrows indicate the direction of time evolution. The horizontal (red) dotted line indicates a typical heating rate limiting these cooling processes. The gray triangles signal the end of the cooling process, defined as the intersection of the curves with the horizontal line accounting for the heating rate.


Fig. 9




Fig. 9 : 

Adapted from [48]: The system's Fermi energy E FS (t ) (left axis) and particle number N S (t ) (right axis). The solid light blue curve is for the Peltier cooling with   and  , γ ev τ 0 =15,  ,  . The dashed-dotted black curve is for evaporative cooling only, with an initial particle number N =N S +N R and ε 1 =1.05E F . Inset: Comparison between the different realizations at short times. The parameters are the same as in Fig. 8.

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