Adiabaticity in the Hamiltonian Functional Integral

D. Sherrington

Edward Farhi, Sam Gutmann

Theoretical Physics

Michel Carreau, Edward Farhi, Sam Gutmann +1 more

Gabriel Cardoso

We study the stability of counterdiabatic drive by computing the joint dependence of the probability of nonadiabatic transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ on the adiabatic parameter $\ensuremath{\delta}$ and on the normalized amplitude of the adiabatic gauge potential (AGP) $\ensuremath{\eta}$ in the Landau-Zener-St\"uckelberg-Majorana model. We show that the Dykhne-Davis-Pechukas formula cannot be readily applied since the AGP introduces a singularity in the Hamiltonian which makes the wave function multivalued in the complex-time plane. This can be understood as the non-Abelian Aharonov-Bohm phase introduced by the AGP and leads to the counterdiabatic correction of the Landau-Zener formula. In particular, it shows that, unlike the nonperturbative suppression of transitions in the adiabatic limit $\ensuremath{\delta}\ensuremath{\rightarrow}0$, the probability is only perturbatively suppressed in the counterdiabatic limit $\ensuremath{\eta}\ensuremath{\rightarrow}1$. We then consider the extension of our results to integrable time-dependent quantum Hamiltonians. We prove that the AGP satisfies the flatness constraint which characterizes integrability in these models, which allows us to derive simple expressions for the AGP and the probability of transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ near adiabatic or counterdiabatic evolution in three- and four-state integrable examples.

Gabriel Cardoso

By the adiabatic theorem, the probability of non-adiabatic transitions in a time-dependent quantum system vanishes in the adiabatic limit. The Landau-Zener (LZ) formula gives the leading functional behavior of the probability close to this limit. On the other hand, in counterdiabatic dynamics, one achieves effectively adiabatic evolution at finite driving speed by adding an extra field which suppresses non-adiabatic transitions: the adiabatic gauge potential (AGP). We investigate the mechanism by which the AGP suppresses the transition probability, changing it from the LZ formula to exactly zero. Quantitatively, we find that adding the AGP to the Hamiltonian modifies the LZ formula by a universal prefactor, independent of the adiabatic parameter, which vanishes in the counterdiabatic regime. Qualitatively, this prefactor can be understood as arising from the Aharonov-Bohm phases generated by the AGP between different paths in the complex time plane. Finally, we show that these results extend to a class of integrable time-dependent quantum Hamiltonians by proving that the AGP preserves their integrability condition.