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arXiv:2312.10664 · The Journal of Chemical Physics

Nonadiabatic transitions during a passage near a critical point

Nikolai A. Sinitsyn, Vijay Ganesh Sadhasivam, Fumika Suzuki

Published Feb 16, 2024• 2 citations
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Abstract

The passage through a critical point of a many-body quantum system leads to abundant nonadiabatic excitations. Here, we explore a regime, in which the critical point is not crossed although the system is passing slowly very close to it. We show that the leading exponent for the excitation probability can then be obtained by standard arguments of the Dykhne formula, but the exponential prefactor is no longer simple and behaves as a power law on the characteristic transition rate. We derive this prefactor for the nonlinear Landau-Zener model by adjusting Dykhne's approach. Then, we introduce an exactly solvable model of the transition near a critical point in the Stark ladder. We derive the number of excitations for it without approximations and find qualitatively similar results for the excitation scaling.

DOI
10.1063/5.0191933

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The passage through a critical point of a many-body quantum system leads to abundant nonadiabatic excitations. Here, we explore a regime, in which the critical point is not crossed although the system is passing slowly very close to it. We show that the leading exponent for the excitation probability can then be obtained by standard arguments of the Dykhne formula, but the exponential prefactor is no longer simple and behaves as a power law on the characteristic transition rate. We derive this prefactor for the nonlinear Landau-Zener model by adjusting Dykhne's approach. Then, we introduce an exactly solvable model of the transition near a critical point in the Stark ladder. We derive the number of excitations for it without approximations and find qualitatively similar results for the excitation scaling.
We examine the effect of nonlinearity at a level crossing on the probability for nonadiabatic transitions P. By using the Dykhne-Davis-Pechukas formula, we derive simple analytic estimates for P for two types of nonlinear crossings. In the first type, the nonlinearity in the detuning appears as a perturbative correction to the dominant linear time dependence. Then appreciable deviations from the Landau-Zener probability PLZ{P}_{\mathrm{LZ}}PLZ​ are found to appear for large couplings only, when P is small; this explains why the Landau-Zener model is often seen to provide more accurate results than expected. In the second type of nonlinearity, called essential nonlinearity, the detuning is proportional to an odd power of time. Then the nonadiabatic probability P is qualitatively and quantitatively different from PLZ{P}_{\mathrm{LZ}}PLZ​ because, on the one hand, it vanishes in an oscillatory manner as the coupling increases, and, on the other, it is much larger than PLZ.{P}_{\mathrm{LZ}}.PLZ​. We suggest experimental situations where the predicted deviations can be observed.
We analyze the coherent quantum evolution of a many-particle system after slowly sweeping a power-law confining potential. The amplitude of the confining potential is varied in time along a power-law ramp such that the many-particle system finally reaches or crosses a critical point. Under this protocol we derive general scaling laws for the density of excitations created during the nonadiabatic sweep of the confining potential. It is found that the mean excitation density follows an algebraic law as a function of the sweeping rate with an exponent that depends on the space-time properties of the potential. We confirm our scaling laws by first order adiabatic calculation and exact results on the Ising quantum chain with a varying transverse field.
We analyse the crossing of a quantum critical point based on exact results for the transverse XY model. In dependence of the change rate of the driving field, the evolution of the ground state is studied while the transverse magnetic field is tuned through the critical point with a linear ramping. The excitation probability is obtained exactly and is compared to previous studies and to the Landau–Zener formula, a long time solution for non-adiabatic transitions in two-level systems. The exact time dependence of the excitations density in the system allows us to identify the adiabatic and diabatic regions during the sweep and to study the mesoscopic fluctuations of the excitations. The effect of white-noise is investigated, where the critical point transmutes into a non-Hermitian 'degenerate region'. Besides an overall increase of the excitations during and at the end of the sweep, the most destructive effect of the noise is the decay of the state purity that is enhanced by the passage through the degenerate region.
We present a comprehensive analysis of the nonlinear Landau-Zener tunneling. We find characteristic scaling or power laws for the critical behavior that occurs as the nonlinear parameter equals to the gap of avoided crossing energy levels. For the nonlinear parameter larger than the energy gap, a closed-form solution is derived for the nonlinear tunneling probability, which is shown to be a good approximation to the exact solution for a wide range of the parameters. Finally, we discuss the experimental realization of the nonlinear model and possible observation of the scaling or power laws using a Bose-Einstein condensate in an accelerating optical lattice.
The adiabatic limit of the transition probability for two-level systems driven by real symmetric time-dependent Hamiltonians is considered. When the Hamiltonian depends analytically on time, the transition probability is governed, in simple cases, by a complex eigenvalue crossing point which is, generically, a square root branching point for the eigenvalues. In such a case, the transition probability is given by the Dykhne formula, the recently discovered geometric prefactor being equal to one. In this paper we deal with the general situation where the relevant eigenvalue crossing point is a branching point of order n/2, n>or=1, for the eigenvalues and a zero of order m>or=0 for the Hamiltonian itself. The analysis shows that the Dykhne formula must be completed by a novel prefactor which depends on both n and m. In particular, this prefactor can take the value zero, in contrast to the geometrical prefactor. We also consider the case where the transition probability is governed by N complex eigenvalue crossing points of different orders n j and m j , j=1, ..., N. The end result displays an interference phenomenon between the individual prefactors similar to the case of generic eigenvalue crossing points n j =1 and m j =0 considered earlier.
It is well known that the dynamics of a quantum system is always nonadiabatic in passage through a quantum critical point and the defect density in the final state following a quench shows a power-law scaling with the rate of quenching. However, we propose here a possible situation where the dynamics of a quantum system in passage across quantum critical regions is adiabatic and the defect density decays exponentially. This is achieved by incorporating additional interactions which lead to quantum critical behavior and gapless phases but do not participate in the time evolution of the system. To illustrate the general argument, we study the defect generation in the quantum critical dynamics of a spin-1/2 anisotropic quantum XY spin chain with three spin interactions and a linearly driven staggered magnetic field.