Transitionless quantum driving for specific eigenstates

Question

Transitionless quantum driving is a concept that was invented by Berry in 2009. In his article on transitionless quantum driving he showed that it is possible to speed up adiabatic evolution of eigenstates without generating transitions between eigenstates. By introducing an auxiliary Hamiltonian known as a counter-diabatic Hamiltonian, it is possible to drive eigenstates of an arbitrary Hamiltonian exactly, that is no transitions occur between eigenstates

The reference to Berry's original article can be found here.

To summarize Berry's article:

By considering a very arbitrary time-dependent Hamiltonian $\hat{H}_0$, with instantaneous eigenstates and eigenenergies given by $$ \hat{H}_0(t)|n(t)\rangle = E_n(t)|n(t)\rangle \tag{1} $$ We have in the adiabatic approximation that the states driven by $\hat{H}_0(t)$ would be $$ |\psi_n(t)\rangle = e^{i(\theta_n(t)+\gamma_n(t))} |n(t)\rangle \tag{2} $$ where $\theta_n(t)$ is the dynamical phase $$ \theta_n(t)=-\frac{1}{\hbar}\int_0^tE_n(s)ds \tag{3} $$ and $\gamma_n(t)$ is the geometrical (Berry) phase $$ \gamma_n(t)=i\int_0^t\langle n(s)|\partial_sn(s)\rangle ds \tag{4} $$ Berry finds an equation such that transition to other eigenstates do not occur. This means that the adiabatic state becomes the exact solution to the Schrödinger equation $$ i\partial_t|\psi_n(t)\rangle=\hat{H}(t)|\psi_n(t)\rangle \tag{5} $$ Applying the time-derivative operator to the adiabatic state $(2)$, we obtain $$ \hat{H}(t) = \sum_n |n\rangle E_n\langle n| +i\hbar\sum_n(|\partial_tn\rangle\langle n|-\langle n|\partial_tn\rangle|n\rangle\langle n|) = \hat{H}_0(t)+\hat{H}_{CD}(t) \tag{6} $$ where I have suppressed the time-dependence for simplicity $|n(t)\rangle\equiv |n\rangle$. $$ \hat{H}_{CD}(t) = i\hbar\sum_n(|\partial_tn\rangle\langle n|-\langle n|\partial_tn\rangle|n\rangle\langle n|) \tag{7} $$ is known as the counter-diabatic Hamiltonian. The sum goes over all the eigenstates satisfying $(1)$. Equation $(6)$ give us the Hamiltonian $\hat{H}(t)$ that drive the eigenstates $|n(t)\rangle$ of $\hat{H}_0(t)$ exactly, even under diabatic-conditions.

Question: In most real experiments, one usually only worries about the evolution of a subset of the Hilbert space, rather than the full Hilbert space.

1. Is it possible to define a Hamiltonian, a state-dependent Hamiltonian, that drive only a specific eigenstate rather then the full Hilbert space of eigenstates as in $(6)$? This specific eigenstate could for example be the ground state.
2. How would it look like?

Answer 1

This answer is probably too late, but Demirplak and Rice developed a similar but slightly more general theory for counter diabatic driving. Demirplak and Rice: On the consistency, extremal, and global properties of counterdiabatic fields (2008). Among other things they show in the article that by writing the Hamiltonian as $$ H(s) = \sum_j E_j(s) P_j(s) $$ where $P_j(s) = |j(s)\rangle\langle j(s)|$ is the projector onto the $j$-th eigenstate, we can find a counter driving Hamiltonian that completely decouples the eigenstate $j$ from the rest of the dynamics. \begin{align*} H_\mathrm{CD}^j(s) = \frac{i\hbar}{T} \left[\frac{\mathrm{d} P_j}{\mathrm{d}s}, P_j(s)\right] \end{align*} It's also noteworthy that in this case the counter driving Hamiltonian is not unique anymore. For any hermitian operator $B$ the operator $$ H_\mathrm{cd}^j + (1 - P_j) B (1 - P_j) $$ will also uncouple the eigenstate. (see Shortcuts to adiabaticity by D. Guéry-Odelin (2019)