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- A system where the adiabatic theorem holds in some cases even for a fast change in the potential energy function.

Suppose some quantum system has a Hamiltonian with explicit time dependence ##\hat{H} := \hat{H}(t)## that comes from a changing potential energy ##V(\mathbf{x},t)##. If the potential energy is changing slowly, i.e. ##\frac{\partial V}{\partial t}## is small for all ##\mathbf{x}## and ##t##, then the adiabatic theorem says that an initial state ##\left|\psi\right.\rangle## at time ##t_1## will be the equivalent state in the basis of new instantaneous eigenstates of ##\hat{H}## at a later time ##t_2##. This is what is called adiabatic passage.

Now, if for example the lowest three instantaneous eigenvalues of ##\hat{H}(t)##, denoted ##E_1 (t)##, ##E_2 (t)## and ##E_3 (t)##, stay constant in time despite the Hamiltonian changing, I would suppose that an initial state superposed from ##\left|\psi_1 \right.\rangle##, ##\left|\psi_2 \right.\rangle## and ##\left|\psi_3 \right.\rangle## at time ##t_1## will be the equivalent state in the new instantaneous eigenbasis at time ##t_2## even if the "passage" is made arbitrarily much faster with scaling ##V(\mathbf{x},t) \rightarrow V(\mathbf{x},\lambda t)## where ##\lambda > 1##.

There seem to be some ways to simulate a faster than normal adiabatic passage even for all eigenvalues non-constant, such as in these references:

https://inspirehep.net/literature/1975583

https://www.nature.com/articles/s41467-021-27900-6

and this has some applications in quantum control theory.

But am I right or missing something here when I assume that an initial state, made of eigenstates of ##\hat{H}## that correspond to time-constant eigenvalues, can change "adiabatically" no matter how fast the change is?

Now, if for example the lowest three instantaneous eigenvalues of ##\hat{H}(t)##, denoted ##E_1 (t)##, ##E_2 (t)## and ##E_3 (t)##, stay constant in time despite the Hamiltonian changing, I would suppose that an initial state superposed from ##\left|\psi_1 \right.\rangle##, ##\left|\psi_2 \right.\rangle## and ##\left|\psi_3 \right.\rangle## at time ##t_1## will be the equivalent state in the new instantaneous eigenbasis at time ##t_2## even if the "passage" is made arbitrarily much faster with scaling ##V(\mathbf{x},t) \rightarrow V(\mathbf{x},\lambda t)## where ##\lambda > 1##.

There seem to be some ways to simulate a faster than normal adiabatic passage even for all eigenvalues non-constant, such as in these references:

https://inspirehep.net/literature/1975583

https://www.nature.com/articles/s41467-021-27900-6

and this has some applications in quantum control theory.

But am I right or missing something here when I assume that an initial state, made of eigenstates of ##\hat{H}## that correspond to time-constant eigenvalues, can change "adiabatically" no matter how fast the change is?

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