11 Corrections to work and efficiency under strong coupling
11.4 Example: Quantum Brownian Motion
Let us finally discuss a simple example to illustrate our findings. We will consider Quan- tum Brownian Motionin the Caldeira-Leggett- or Ullersma-model [64, 65]. It consists of a single central harmonic oscillator which is linearly coupled to a bath ofn harmonic os- cillators. The spectral density of the bath is assumed to be Ohmic with some cut-off ωmax.
The total Hamiltonian takes the form
H(0) =H(0)S+gV(g) +HB+HL, (11.77) with H(0)S = 1 2mω 2x2+ p2 2m, (11.78) HB=
∑
k 1 2mkω 2 kx2k+ p2k 2mk , (11.79) V(g) =x∑
k gkxk+HL, (11.80) HL =gx2∑
k g2k mkω2k, (11.81)where the Lamb-shift HL renormalizes the system’s oscillator frequency in presence of
the interactions to the bath. It ensures that the Hamiltonian remains bounded from below and that the system thermalizes to the HamiltonianH(0)Sin the Markovian dynamics that
A Q UA N T U M O F T H E R M O DY N A M I C S 149 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00 0.05 0.10 0.15 0.20 0.25 0.30 g Work 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 g Power×104
Figure 11.1: Left figure: The extracted work is plotted versus interaction strengthg. The blue dots show the exact extracted work by computing the unitary evolution ofSB for a protocol which becomes optimal in the weak coupling regime. The solid, light blue line shows the corresponding predictions using our framework. The dashed blue line shows the maximum extractable work as a function ofg. The dashed grey line shows the maximum extractable work in the weak coupling limitW(weak). Parameters are given by ω=1, mk=m=1, ωmax=1.2 and β=3.5, βS=1. The bath consists ofn=165 oscillators. The protocol consists of 200 quenches, with a waiting time 10/g2when computing the unitary dynamics. For values ofg smaller than the ones shown larger bath sizes are required for proper thermalization of the system and lead to larger errors for the bath considered here. Right figure: Plot of power versus interaction strength. The blue dots again show the exact results obtained from unitary evolution and the solid, light blue line the effective description using our framework. The parameters are the same as in the left figure.
results from taking the limit of weak interactions if the bath is thermal. The ohmic spectral density means that the frequencies ωkand couplingsgktake the values
ωk= k nωmax, gk=ωk r 2ωmax πn . (11.82)
This choice ensures that the spectral density
J(ω):= 2 π
∑
kg2k
ωk
δ(ω−ωk) (11.83)
approaches a linear functionJ(ω)∝ ω with cut-off ωmaxin the continuum limitN→∞.
This model is an important in the field of open quantum systems [277], in particular because it is both exactly solvable and efficiently simulable on a computer, because it is quadratic in the bosonic annihilation and creation operators (see, e.g., Refs. [278, 279]). Therefore it is also a good test-bed in quantum thermodynamics and has found numerous applications there-in (see, e.g., Refs. [256, 258, 261, 267, 269, 270, 280–282]).
Importantly, this model is known to have the property that the state onS equilibrates to the reduced state of the thermal state of the full system in the limit of large baths indepen- dently of the coupling strength and hence fulfills our basic assumption of thermalization after a single quench [283, 284]. Note however, that this does not automatically imply that the effective description in terms of iterated Gibbs-states is valid. Furthermore, the interaction is not of the formgV, due to the presence of the Lamb-shift. It formally hence does not fit our framework developed in the previous sections. It can easily be seen from the derivations, however, that to leading order in g, this latter discrepancy does not make a difference. In particular, to leading order the optimal Hamiltonian local Hamiltonians in work-extraction protocols are again determined by the renormalized interaction ˜V, which is now given by ˜ V=x
∑
k gkxk−xTr ωβ(HB)∑
k gkxk ! =x∑
k gkxk, (11.84)which follows from the reflection-symmetry of the harmonic-oscillator. This result implies that, to leading order ing, the optimal work-extraction protocols are identical to the weak- coupling protocols.
Let us now discuss work-extraction protocols in the Caldeira-Leggett model. We assume an initial state ωβS(HS) ⊗ωβ(HB)with βS 6=β. In the weak coupling limit, the optimal
m 7→ βS/β m. Note that often harmonic oscillators appear as effective desciptions in
which the parameterm does not correspond to the actual, physical mass of some particle and can be modified. Not being able to change the parameterm will naturally result in protocols that are not optimal. This first quench is then followed by an isothermal process back to the initial Hamiltonian. This is nicely illustrated in Fig. 11.1, where we calculated numerically the results obtained from exact unitary time-evolution and the predictions in our model, showing good agreement. In particular, the extracted work decreases asg2with
the coupling strength, as expected from our general results.
Turning to the power of this protocol, we then consider the protocol with a fixed num- ber of steps N and study the power as a function of g. We numerically found that the equilibration time in this model is proportional to1/g2(forg<1), in agreement with our considerations in the previous sections. Hence, we expect that the power shows a behaviour of the formP(g)∝ W(weak)g2−O(g3). This behaviour can indeed be seen in Fig. 11.1 as expected. The example thus shows good agreement with our predictions.
11.5
Summary
Standard thermodynamic bounds usually hold for systems that are weakly coupled. This weak coupling behaviour can be well justified for macroscopic systems which interact lo- cally, due to the vanishing surface-to-volume ratio. For truly small systems this argument does not hold and we hence have to study more specifically in how far thermodynamic bounds remain valid or have to be corrected. In this chapter, we have derived fully gen- eral expressions for the strong-coupling corrections of a thermal machine coupled to a heat bath. To arrive at our results, we employed the framework developed in the previous chap- ter. We have explicitly shown that the corrections to the weak coupling bounds become irrelevant for macropscopic systems. Our expressions are completely general and only rely on the assumption that the systems in questions actually thermalize in the sense of closed systems when coupled to a heat bath. Interestingly, the correction terms can be expressed succinctly in terms of relative entropies, and essentially measure irreversible dissipation in terms of correlations which are caused from the interaction between the system and bath.
In the case of weak, but finite coupling strength, we derived the explicit leading correc- tions to work-extraction bounds and the efficiency of optimal thermal machines as a func- tion of the system and bath Hamiltonians and their interaction operator. The leading terms are at second order in the interaction strength and take a fairly simple form. The fact that the corrections are of second order in the coupling strength may not come as a surprise, since it can be argued from general grounds: Assuming that the universal weak-coupling bounds also apply for strongly coupled machines (which is necessary for thermodynamic consis- tency and matches our results) and that optimal work-extraction bounds vary smoothly ing already implies that corrections have to be at second order without making any calculation. The merit of the results in this chapter is thus that they provide the explicit functions which can in principle be evaluated for any model.
The results in this chapter show that, in terms of efficiency of thermal machines, finite coupling strength are detrimental when compared to the weak-coupling limit. We also discussed that our results imply that the power vanishes in the limit of arbitrarily strong interactions and weak interactions. Hence optimal power is given by a finite coupling strength as expected.
All our results can be demonstrated in the paradigmatic example of Quantum Brownian motion, which is an integrable (exactly solvable) model. It would be interesting to study the explicit corrections for models which are fully interacting. In this chapter, we had to assume that the thermal machine never interacts with both heat baths at the same time. An interesting avenue for further research would be to study autonomous thermal machines in the strong coupling regime, which are simultaneously coupled to both thermal baths and find out whether similar bounds can be derived.