So far, we have treated only thermal contacts to a single heat bath. From now on will consider the possibility to put the system in contact with heat baths at two different in- verse temperatures βcand βh with βc > βhand study machines which extract work by
alternatingly combining adiabatic evolution and thermal contact to one of the heat baths. Furthermore, we will consider cyclic protocols – where now a cyclic protocol means that after a given number of steps of the protocol the system returns to both its initial state and Hamiltonian, whereas previously only the Hamiltonian returned to its initial condition. Note that in such a setting, a protocolP is independent of initial conditions: any pair of state and Hamiltonian(ρt, Ht)during the protocol returns to its initial state after one cycle
of the protocol.
Let us first make some general observations about any such protocol. In particular, we will be interested in the efficiency of the process, which we define in the traditional way as
η(P ):= W(P )
Qh(P )
, (9.25)
whereW(P )is the total work extracted during one cycle andQh(P )is the total heat ab-
sorbed from the hot reservoir. Unsurprisingly, we obtain Carnot’s bound for the efficiency for any cyclic protocol:
Theorem 9.4 (Carnot efficiency). For any cyclic protocol P between two heat baths at inverse temperatures βc>βh>0 the efficiency fulfills
η(P ) ≤ηc:=1− βh
βc.
Furthermore, if the set of HamiltoniansHis unrestricted, the bound can be saturated. To prove this theorem, we need to first introduce the notion of a Carnot-like protocol. Let us assume that the protocol starts with the system in the thermal state of Hamiltonian H1at the temperature of the hot heat bath. Then such a protocol consists of four parts:
1. An isothermal in contact with the hot bath from HamiltonianH1to HamiltonianH2,
2. an adiabatic evolution from HamiltonianH2to HamiltonianH3, implementing a unitary
transformationU,
3. an isothermal in contact with the cold bath from HamiltonianH3to HamiltonianH4,
4. an adiabatic evolution back fromH4to HamiltonianH1, implementing a unitary trans-
formationV.
Importantly, the only dissipation in such a protocol can occur when the system is put in thermal contact with one of the heat baths after the adiabatic evolution. The condition for this dissipation to vanish is that the final state of the adiabatic evolution matches the thermal state at the beginning of the isothermal. Thus, there is no dissipation if and only if
Uωβh(H2)U †=
ωβc(H3) and Vωβc(H4)V †=
ωβh(H1). (9.27)
Due to the close correspondence between Hamiltonians and thermal states, we can express this condition as UH2U†= βc βh H3 and V H4V†= βh βc H1. (9.28)
The heat exchanged with the bath in the first isothermal is given by
Qh(P ) =Th∆S1→2h :=Th(S(ωβh(H1)) −S(ωβh(H2))). (9.29)
Similarly, the heat exchanged during the isothermal in contact with the cold bath is given by
Qc(P ) =Tc∆S3→4c = −Tc∆S1→2h = −
Tc
Th
Qh(P ), (9.30)
where we have used that the whole protocol is cyclic and the total change of entropy van- ishes along one cycle. Using the first law of thermodynamics we then obtain the efficiency of the protocol as η(P ) = W(P ) Qh(P ) = Qh(P ) +Qc(P ) Qh(P ) =1− Tc Th . (9.31)
To see that no protocol can exceed Carnot efficiency, observe that any protocol can be sub- divided into segments in which only thermal contacts to the hot bath occur and segments in which only thermal contacts with the cold bath occur. If these segments are not isothermal processes, the extracted work will be decreased and the heat will be increased. At the same time, the matching conditions (9.28) will not be matched, with a similar effect. Therefore the efficiency will be reduced in comparison to the Carnot efficiency.
Given the above result, it is a natural question to ask whether the Carnot-bound can also be achieved with only limited control over the Hamiltonians. For example, we might consider as working medium an interacting spin system where the experimenter can only control external magnetic fields. It is then interesting to know whether Carnot efficiency can in principle be achieved, and whether this depends on further properties of the model. We will find in section 9.6 that the achievable indeed strongly depends on the sign and strength of the interaction in such an example.
Before we come to that example, let us see from a more abstract point of view what could go wrong. From the above discussion, we see that we need to be able to implement isothermals and meet the matching conditions (9.28) to achieve Carnot efficiency. Since we always assume that the set of allowed HamiltoniansHis well-behaved, in particular any
A Q UA N T U M O F T H E R M O DY N A M I C S 107
two Hamiltonians can be connected by a smooth curve, isothermal processes are always possible. The matching condition, however, cannot always be met, in the same way as for a single heat bath in Lemma 9.2. The following theorem then gives a similar bound as Lemma 9.2 for the case of efficiencies of thermal machines.
Theorem 9.5 (Efficiency under control limitations). LetPbe any cyclic protocol between two heat baths at inverse temperatures βc>βh>0 employing Hamiltonians from the set
Hand fully thermalizing mapsTtas thermal contacts. Then there exist four Hamiltonians
H1, . . . , H4∈ Hsuch that the efficiency is bounded as
η(P ) ≤1− βh βc ∆S+minUD(Uωβh(H2)U †kω βc(H3)) ∆S−minVD(Vωβc(H4)V†kωβh(H1)) , (9.32) where U, V are unitaries that can be generated by time-dependent Hamiltonians inH, ∆S=S(ωβh(H2)) −S(ωβc(H4))andS the von Neumann entropy. Furthermore, for any
choice ofH1, . . . , H4∈ Hthere exists a protocol that saturates the bound.
The formal proof of the theorem is given in section 14.6.6. It roughly works in the same way as theorem 9.4, but taking care of the dissipation terms. The theorem gives a tight, but very abstract bound. In particular, at first sight the optimization over all the unitaries that can be implemented using time-dependent Hamiltonians formHposes a serious challenge to evaluating it. In the next section, we will see that in generic situations this optimization can in fact be eliminated.