Integrable time-dependent Hamiltonians, solvable Landau–Zener models and Gaudin magnets

Abstract

We solve the non-stationary Schrödinger equation for several time-dependent Hamiltonians, such as the BCS Hamiltonian with an interaction strength inversely proportional to time, periodically driven BCS and linearly driven inhomogeneous Dicke models as well as various multi-level Landau-Zener tunneling models. The latter are Demkov-Osherov, bow-tie, and generalized bow-tie models. We show that these Landau-Zener problems and their certain interacting many-body generalizations map to Gaudin magnets in a magnetic field. Moreover, we demonstrate that the time-dependent Schrödinger equation for the above models has a similar structure and is integrable with a similar technique as Knizhnikov-Zamolodchikov equations. We also discuss applications of our results to the problem of molecular production in an atomic Fermi gas swept through a Feshbach resonance and to the evaluation of the Landau-Zener transition probabilities.

Quantum Annealing and Thermalization: Insights from Integrability

“…(19) with numerical results that were obtained by direct calculations of all microstate probabilities and then using Eq. (18). Figure 5(a) shows that total entropy of the final distribution saturates at a finite value in the limit N → ∞, which is consistent with our result in the main text that the number of errors is restricted to O(N 0 ) values in this limit.…”

Introduction to the Dicke Model: From Equilibrium to Nonequilibrium, and Vice Versa

“…The existence of such disorder prevents the simplification of replacing individual spins by a collective spin operator, hence the need for efficient numerical methods to explore this enlarged Hilbert space . It is however notable that in the case where is disordered, while , the model can be shown to be integrable, as a special case of a Richardson–Gaudin model . In addition to the dynamics, one can also calculate the phase diagram of the disordered Dicke model by mean‐field approaches, showing that the disorder does not destroy the superradiant phase, but modifies the phase boundary.…”

“…[134] It is however notable that in the case where ω j z is disordered, while λ j = λ, the model can be shown to be integrable, as a special case of a Richardson-Gaudin model. [135][136][137][138] In addition to the dynamics, one can also calculate the phase diagram of the disordered Dicke model by mean-field approaches, [128,129] showing that the disorder does not destroy the superradiant phase, but modifies the phase boundary.…”

Counterdiabatic Hamiltonians for multistate Landau-Zener problem

“…Our method is general and is applicable to the general Hamiltonian which is not necessarily solvable. The previous studies treated solvable Hamiltonians and the physical meaning of the auxiliary Hamiltonians was not clear [22][23][24]. The present work is useful not only to clarify the physical meaning of the auxiliary Hamiltonians but also to calculate the transition probabilities for general time-dependent systems.…”

“…The auxiliary Hamiltonian plays a role of the generator for the corresponding parameter. Several solvable models were treated to construct the auxiliary Hamiltonians and to demonstrate that the method is useful to describe the time evolution [22][23][24].…”