**2. Weak Invariant and Isoenergetic Process**

The concept of weak invariants has wide universality. Let us consider a master equation

$$\text{id}\,\frac{\partial}{\partial t}\frac{\rho}{t} = \text{\textquotedblleft}\rho\text{)}\tag{1}$$

for the density matrix ρ describing the state of a quantum subsystem, where £ is a certain linear superoperator. Henceforth, is set equal to unity for the sake of simplicity. Then, a weak invariant *I* = *I*(*t*) associated with this master equation is defined as a solution of the following equation [7]:

$$i\frac{\partial I}{\partial t} + \underline{\mathsf{E}}^\*(l) = 0.\tag{2}$$

In this equation, £\* stands for the adjoint of £ defined by tr (*Q* £(*R*)) = tr(£*\** (*Q*) *R*), provided that *Q* £(*R*) and therefore also £*\** (*Q*) *R*] should be trace-class. Then, it follows from Equations (1) and (2) that the expectation value of *I* is conserved:

$$\frac{d\langle I\rangle}{dt} = 0\tag{3}$$

with the notation *Q* ≡ tr(*Q*ρ), although the spectrum of *I* is time-dependent, in general.

Our purpose is to clarify the implication of the energy bath to quantum thermodynamics. For it, what to be contemplated is the isoenergetic processes, along which the internal energy of the subsystem is kept constant through energy transfer between the subsystem and the energy bath. A point is that the energy transfer that may be realized not in the form of heat but rather in the dynamical manner [8,9].

The isoenergetic processes have been discussed for a "three-stroke" engine [3,10], in which however no explicit time evolution of the subsystem has been considered. In view of finite-time quantum thermodynamics, a density matrix evolves in time according to a master equation. Here, let us assume Markovianity of the subdynamics. In this case, the superoperator in Equation (1) has the Lindblad form [4,5]:

$$\hat{\mathsf{E}}(\rho) = [H(\mathsf{t}), \ \rho] - \frac{i}{2} \sum\_{i} a\_{i} \Big( \mathrm{L}\_{i}^{\dagger} \, \mathrm{L}\_{i} \, \rho + \rho \, \mathrm{L}\_{i}^{\dagger} \, \mathrm{L}\_{i} - 2 \mathrm{L}\_{i} \rho \mathrm{L}\_{i}^{\dagger} \Big), \tag{4}$$

where *H*(*t*) is the time-dependent Hamiltonian and *a <sup>i</sup>*'s are nonnegative *c*-number coefficients. Both the coefficients and the Lindbladian operators *L <sup>i</sup>*'s may also depend on time, explicitly.

The weak invariant relevant to the isoenergetic process is the time-dependent Hamiltonian itself. Thus, the internal energy

$$\mathcal{U} = \text{tr}(H(t)\,\rho) \tag{5}$$

is required to be conserved under time evolution generated by the Lindblad equation. Such a condition is fulfilled if the Hamiltonian obeys the following equation:

$$\frac{\partial}{\partial t} \frac{H(t)}{t} - \frac{1}{2} \sum\_{i} a\_i \left( L\_i^\dagger \frac{\dagger}{i} L\_i \, H(t) + H(t) \, L\_i^\dagger \, L\_i - 2L\_i^\dagger \frac{\dagger}{i} H(t) \, L\_i \right) = 0,\tag{6}$$

which comes from Equation (2) with Equation (4).

In general, it is a nontrivial task to determine the Lindbladian operators in Equations (4) and (6), since it needs detailed knowledge of how the subsystem interacts with the environment. However, Equation (6) tends to make it possible to determine them without such knowledge [11].

Closing this section, we make some comments on the concept of weak invariants. Firstly, it has recently been discovered [12] that there exists a connection between weak invariants and the action principle for corresponding master equations. Secondly, such quantities clearly have their classical analogs. For instance, a weak invariant associated with the classical Fokker-Planck equation is discussed in Reference [13]. Thirdly, it is worth mentioning that the Lindblad equation with a time-dependent Hamiltonian can describe the dynamics toward an "instantaneous attractor" in the nonadiabatic regime [14], inside which isoenergeticity is realized.
