4.4 Implications of Negative Absolute Temperatures
4.4.2 Carnot Efficiency
The Carnot efficiency indicates the largest efficiency achievable in a closed cycle of a working body operating between a hot and a cold reservoir at temperaturesTHandTC, respectively.
Such a cycle operating at the Carnot limit is characterized by zero entropy production.
The efficiency η of a heat engine is given by the ratio of work W performed during one cycle over the heatQH that is extracted from the hot reservoir,
η= W QH
= QH−QC
QH
= 1−QC
QH
= 1−TC
TH
, (4.20)
whereQC is the heat dumped into the cold reservoir. In the last step, the relationδQ= TdS for reversible processes was used where dS is the entropy change. Equation4.20 is also applicable to negative temperatures. The results, however, may sound counterintuitive at first sight and have to be clarified.
Entropy
S
S
QH QC
Entropy
S
S QC QH
Energy
Entropy
S S
QH QC
A
B
C
QC QH
S S
W
TC > 0 TH > 0
reservoircold hot
reservoir working
body
QC QH
S S
W
TC < 0 TH < 0
QC QH
S S
W
TC > 0 TH < 0
Figure 4.5: Illustration of constant entropy machines. Heat (QH,QC), work (W) and entropy flows (S) are indicated by arrows, where the length of the arrows in the entropy versus energy graphs on the left is exaggerated. In an ideal Carnot cycle, the reservoirs are infinitely large such that the temperatures TH and TC are constant throughout the cycle and the arrows in the diagram are infinitesimal. In the schematic diagrams of the cycles on the right, the width of the arrows qualitatively indicates the amount of heat or work. A, The usual Carnot cycle operating between two positive temperature reservoirs leads to an efficiency between 0 and 1. B, If both reservoirs are at negative temperature, the efficiency is negative and therefore work must be supplied to maintain the cycle. This cycle is the mirror image of the case with positive temperatures. CIf the hot reservoir is at negative temperature and the cold reservoir at positive temperature, the efficiency is larger than 1. In this case, heat can be extracted from both reservoirs simultaneously.
In the case of both temperatures TH and TC being positive (Fig. 4.5A), the resulting efficiency is always between 0 and 1, i.e. some fraction of the extracted heat from the hot reservoir is converted into work, and the remainder is dumped into the cold reservoir such that the increase in entropyS of the cold reservoir precisely balances the decrease in
4.4 Implications of Negative Absolute Temperatures
entropy of the hot reservoir and the net entropy production during the cycle is zero. If the heat engine is reversed, work must be supplied to maintain the cycle. The cycle then operates as a heat pump.
If both reservoirs are at negative temperatures (Fig.4.5B), thenTC/TH=|TC|/|TH|>1.
Therefore, the efficiency is negative,η <0, and its absolute value can be very large. Thus, work must be performed to transfer heat from the hotter to the colder reservoir, in contrast to positive temperatures. In this case, entropy flows from the cold to the hot reservoir, also in contrast to the positive temperature case. If the engine is reversed, it delivers work by transporting heat from the cold to the hot reservoir [2], which is not possible in the case of positive temperatures. The roles of engine and heat pumps are thus reversed compared to the case with positive temperatures above: The cycle between negative temperatures is the mirror image of the cycle between positive temperatures.
A peculiar case is given if the hot reservoir is at negative temperature and the cold reservoir at positive temperature. The ratio of the temperatures is negative in this case, TC/TH<0, leading to a Carnot efficiency above unity,η >1. Thus, the work delivered by the engine is larger than the heat extracted from thehot reservoir. At first sight, this may seem like a contradiction to energy conservation. However, the key for an understanding of this seemingly unphysical result lies in the slope of the entropy curve (Fig. 4.5C). In contrast to positive temperatures, at negative temperatures entropy increases when heat is extracted from a reservoir. Thus, heat can be extracted from both reservoirs at the same time, while the increase of entropy of the negative temperature reservoir is precisely compensated by a decrease of entropy of the positive temperature reservoir. The work produced by the cycle is therefore, instead of the difference as in the usual case, the sum of both heat quanta,W =QH+QC. Also in this case, entropy flows from the cold to the hot reservoir, in contrast to the usual heat engine at positive temperatures. If this engine is reversed, work must be supplied to increase the energy in both reservoirs simultaneously.
To realize such an engine, however, one would need to find a way to couple these two reservoirs. It turns out that the regions of opposite temperature are not adiabatically connected atβ= 0, i.e. it is not possible to drive a system across the planeβ= 0 without producing entropy [157]. Therefore a Carnot engine that operates between a positive and a negative temperature reservoir throughβ= 0 is impossible. Still, it is in principle possible to build an engine that operates between a positive and a negative temperature reservoir where the efficiency, due to the entropy production when crossing β = 0, is below the Carnot limit but still above unity. Alternatively, the ideal Carnot efficiency can be reached ifβ= 0 is not crossed quasi-statically but if a process connects the eigenstates at positive and negative temperature exactly such that no entropy is produced. For example, an ideal π-pulse inverts the population of a two-level system without producing entropy.
Figure 4.6 shows a possible realization of an engine with efficiency above unity in the Carnot sense. It consists of three stable two-level systems as heat reservoirs and working body, e.g. hyperfine states of atoms with negligible spontaneous emission that thermal-ize via collisions. The working body is initially at very low temperature TW,i ≈0 where
the ground state is predominantly occupied. When the working body is coupled to the cold reservoir at positive temperatureTC> TW,i, heat and entropy are transferred to the working body untilTW=TC, i.e. the occupation of the excited state of the working body increases. After decoupling, the working body is coupled to the hot reservoir at tempera-tureTH≈ −0. Again, heat flows into the working body untilTW=TH, corresponding to a strong occupation of the excited state. This heat flow is accompanied by a reverse entropy flow into the hot reservoir. During both couplings, also some entropy is created as these couplings are nonadiabatic processes. After decoupling, a coherent external electromag-netic field is applied to extract the energy from the working body. A π-pulse of the field leads to a de-population of the excited state until TW ≈0, similarly to the initial state;
simultaneously the field is amplified. Thus, work is extracted from the working body which was previously gained from both the cold and the hot reservoirs, leading to an efficiency η >1.
QC QH
SC SH
W
TC > 0 TH ≈ -0
cold reservoir working body hot reservoir
�-pulse TW
1 2
3
Figure 4.6: Possible realization of a heat engine with efficiencyη >1. A working body operates between a cold reservoir at positive temperature TCand a hot reservoir at negative temperatureTH, where all three systems consist of stable two-level systems. The height of the bars schematically indicates the occupation probabilities of the ground and the excited state of the reservoirs and of the initial state of the working body. In steps 1 and 2, the working body is consecutively brought into thermal contact with the two reservoirs and the heat QCand QHflows during the subsequent thermalization. The corresponding entropy flows are indicated bySCandSH. In step 3, work is performed by the working body on an external electromagnetic field, which is applied as aπ-pulse. For details, see main text.