6.5. Marco conceptual
6.5.5. Visualización y abstracción matemática
For the Gibbs equilibration model and the time-average equilibration model, we can check whether the minimum work principle is valid in a given thermodynamic protocol by look- ing at whether the final state in the quasi-static realisation is passive. This is possible in the Gibbs equilibration model due to the close link between energy, entropy and tempera- ture and in the case of the time-average model due to its close connection to passivity and random unitary quantum channels. In the case of arbitrary GGEs both these links are not available to us. It is therefore not surprising that we won’t be able to formulate general statements in the sense of the previous two sections. In this section, we will therefore dis- cuss numerical examples using free Fermions instead. In particular, we will see examples for the following behaviours:
1. Initial conditions and a Hamiltonian trajectory where the minimum work principle re- mains valid.
2. Initial conditions and a Hamiltonian trajectory where the minimum work principle fails, even though the effective description in terms of Gibbs states would suggest that it is valid.
We will further discuss in detail how these behaviours can be understood and connect them, in the spirit of the last two sections, to the concept of passivity. The reasons for discussing free fermionic systems plentiful:
i) They are known to be well described by GGEs, but not by Gibbs states, due to their integrability [40, 42].
ii) They can be efficiently simulated on a computer, even for large system sizes.
iii) They can be simulated with ultra-cold atom in optical lattices, making our predictions testable in experiments [170, 173, 253, 254].
While focussing on free fermionic systems here, we can expect that similar results can be derived for free bosonic systems as well, which have a very similar structure. Here, the aim
A Q UA N T U M O F T H E R M O DY N A M I C S 131
is primarily to present examples that show different behaviour, including a violation of the minimum work principle, and which can be understood conceptually. For more detailed numerical investigation that appeared as follow-up to the work presented in this chapter see Ref. [255].
In the following, we will consider a simple one-dimensional chain ofn free fermions, with an initial Hamiltonian of the form
H(0) = n
∑
j=1 ejfj†fj+g n−1∑
j=1 (fj†fj+1+fj+1† fj), (10.57)where the ejdenote on-site potentials andg determines the amplitude for hopping between
neighboring sites. Any such Hamiltonian can be brought into the normal-form
H(0) = n
∑
k=1 µkηk†ηk = n∑
k=1 µknk, (10.58)where ηk, η†kare again fermionic operators that are related to thefjby a canonical transfor-
mation (an×n unitary matrix). The number operators nk mutually commute and hence
also commute with the Hamiltonian H(0). They provide the relevant set of conserved quantities in the case of initial states that are Gaussian. For a more detailed discussion of this point see Ref. [249]. Gaussian states are those states that are fully determined by the correlation functionsC(ρ)ij:=Tr(ρ fifj)through Wick’s theorem and include eigenstates
and Gibbs-states of quadratic Hamiltonians such asH(0). Importantly, if we consider the GGE with the conserved quantitiesnkit takes the form
ωGGE(ρ, H(0),{nk}) =
e−∑ λknk
Z . (10.59)
It therefore can be understood as a Gibbs state of the quadratic Hamiltonian∑kλknkand is
also a Gaussian state. This is even the case if the state ρ from which the GGE is determined is not Gaussian. This is well in accordance with recent results that show that a non-Gaussian initial state evolves towards a Gaussian one under the dynamics of a short-ranged quadratic Hamiltonian [189, 196]. Furthermore, it implies that we can completely focus our attention to Gaussian states even if the initial state of the thermodynamic protocol is not Gaussian, since nothing changes on the level of the effective description if we replace this initial state with the Gaussian state which has the same correlation functions C(ρ)ij. This is
even true on the level of exact, time-dependent, unitary dynamics, since time-evolution under quadratic Hamiltonians maps Gaussian states to Gaussian states and can be expressed solely on the level of the correlation matrices in closed form.
Since we can restrict to Gaussian states, all the analysis of thermodynamic protocols can be reduced to the level of correlation matricesC(ρ). These aren×n-matrices in contrast
to the full density matrices of size2n×2n, which enables us to compute the full, unitary time-evolution of system and bath together in an efficient manner and compare it with the effective description.
In the following I will discuss two exemplary cases of work-extraction protocols from different initial conditions. To do that let us split the HamiltonianH(0)into three parts: a system Hamiltonian on the first siteHS = e1f1†f1, an interaction termV = g(f1†f2+
f2†f2)and the bath Hamiltonian HB = H−HS−V. We now assume that we can only
change the Hamiltonian by adjusting the potential e1on the first site and that ej =e is fixed
for allj 6=1. The two examples will be optimal work-extraction protocols from different initial conditions, leading to completely different behaviour. Before discussing the results in the specific examples, let us first discuss optimal work-extraction protocols in the case of free fermionic systems with GGEs as effective description.
As discussed above, the whole analysis can be carried out on the level of correlation matrices. On the level of correlation matrices, going from a state ρ to its associated Gener- alized Gibbs ensemble corresponds to projecting the correlation matrixC(ρ)to its diagonal
in the basis of the normal-modes ηk. It is therefore very similar from a formal point of view
to the case of the time-average equilibration model on the level of density matrices. To make this clearer, let us call a correlation matrixC passive with respect to the Hamiltonian
0.5 0.4 0.3 0.2 0.1 0.0 5 10 15 20 25 30 Work N 0.6
Figure 10.2: Extracted work with quenches only on a single site of the chain of Fermions in the first example. Dark blue correspond to the work W computed from the exact unitary evolution, light blue points to the work WGGEcomputed from the effective description in terms GGE states, and red points to the effective description using Gibbs states. As an initial state we take, β=1/2, Tr(f†
1f1ρ(0)S) =0.1, n=100. For the initial Hamiltonian, e0=0.1, ei=1∀i6=1, g=0.5. The protocol consists of a first quench to e1=4.3, followed by N−1 equidistant quenches back to the original Hamiltonian. The exact evolution is obtained by letting system and bath interact for a random time between20/g and 100/g much larger than the equilibration time. (Figure adapted from Ref. [3].)
H=∑kµknk, if it is diagonal in the basis of the normal-modes and the populations of the
normal modes decrease with increasing energy of the mode:
Ckk≥Cll =⇒ µk ≤µl. (10.60)
It then follows by essentially the same arguments as in the time-average equilibration mod- els that optimal protocols have the property that the final correlation matrix has the same spectrum as the initial correlation matrix and is passive. Similarly, the minimum work principle holds in the case that the final correlation matrix in the reversible limit is passive. Not again however, that optimal protocols in general require being able to quench to arbitrary free fermionic Hamiltonians and hence one usually cannot expect to reach this bound in the system-bath setting. Similarly, in the system-bath setting it might be impossi- ble to find protocols with final states that have a passive correlation matrix.
A further point that will become important later is that a passive correlation matrix does not imply that the state from which it is calculated is passive as a quantum state with respect to the Hamiltonian on the full many-particle Hilbert-space. A simple example of such behaviour is discussed in section 14.7.1. Let us now come to the specific examples.