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arXiv:2506.11759 · Europhysics Letters (EPL)

Suppressing excitations in the nonlinear Landau-Zener model

Sebastian Deffner, Steve Campbell

Published Aug 20, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Many complex quantum systems can be described by effectively nonlinear dynamics. While such dynamics have many appealing characteristics, they also make the analysis significantly more involved. This is due to the fact that only a few analytical treatments exist, and that the language of quantum mechanics is built for linear operators. For instance, the very formulation of the quantum adiabatic theorem requires the underlying dynamics to be linear. In this work we show that in a generalized Landau-Zener model, nonlinear dynamics can be leveraged to suppress excitations and coherences of the corresponding linear scenario. To this end, we introduce a generalized “energy spectrum”, which is defined by the expectation values of the energy under the stationary states. As a main result, we show that the nonlinear term in the evolution equation acts like an effective shortcut to adiabaticity for the linear Landau-Zener problem.

DOI
10.1209/0295-5075/adfd7a

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•Jan 29, 2009•0 citations
Abstract Many complex quantum systems can be described by effectively nonlinear dynamics. While such dynamics have many appealing characteristics, they also make the analysis significantly more involved. This is due to the fact that only a few analytical treatments exist, and that the language of quantum mechanics is built for linear operators. For instance, the very formulation of the quantum adiabatic theorem requires the underlying dynamics to be linear. In this work we show that in a generalized Landau-Zener model, nonlinear dynamics can be leveraged to suppress excitations and coherences of the corresponding linear scenario. To this end, we introduce a generalized “energy spectrum”, which is defined by the expectation values of the energy under the stationary states. As a main result, we show that the nonlinear term in the evolution equation acts like an effective shortcut to adiabaticity for the linear Landau-Zener problem.
We develop a general theory of Landau-Zener (LZ) tunneling in a two-level system with amplitude-dependent, sign-reversible nonlinear coupling, distinguishing it fundamentally from conventional on-site nonlinearity. Through a combination of analytical and phase-space analysis, we show that beyond a critical interaction strength, the nonlinear coupling fundamentally reshapes the adiabatic energy landscape, introducing a topological twisted and knotted structure. This structure leads to a complete breakdown of the standard exponential LZ formula, even in the adiabatic limit. Central to this anomalous behavior is the emergence of a black-hole-like fixed point, which acts as a universal attractor: upon traversing the critical region, all quantum trajectories converge to this fixed point, irreversibly erasing any memory of the initial state. From this fixed-point picture, we derive an exact analytical expression for the adiabatic tunneling probability, revealing a characteristic power-law dependence on both linear and nonlinear coupling strength. Our work establishes a paradigmatic framework for nonlinear-coupling-induced anomalous adiabaticity breaking and offers a universal mechanism for state control in driven quantum and wave systems.
Quantum adiabatic dynamics is the crucial element of adiabatic quantum computing and quantum annealing. Shortcuts to adiabaticity enable acceleration of the computational time by suppressing unwanted non-adiabatic processes with designed classical fields. Here, we consider quantum state transfer in the Landau-Zener model, which exemplifies the key elements of quantum adiabatic dynamics. We argue that non-adiabatic transitions can be suppressed by autonomous quantum dynamics, which involves coupling the Landau-Zener qubit to a second quantum system. By tuning the coupling strength, the composite quantum dynamics can reduce the probability of unwanted processes by more than two orders of magnitude. This is a prime example of control where the quantum properties of the control fields are key for implementing shortcuts to adiabaticity.
We investigate the Landau-Zener tunneling (LZT) of a self-interacting two-level system in which the coupling between the levels is nonreciprocal. In such a non-Hermitian system, when the energy bias between two levels is adjusted very slowly, i.e., in the adiabatic limit, we find that a quantum state can still closely follow the eigenstate solution until it encounters the exceptional points (EPs) at which two eigenvalues and their corresponding eigenvectors coalesce. In the absence of the nonlinear self-interaction, we can obtain explicit expressions for the eigenvectors and eigenvalues and analytically derive the adiabatic LZT probability from invariants at EPs. In the presence of the nonlinear interaction, the dynamics of the adiabatic evolutions are explicitly demonstrated with the help of classical trajectories in the plane of the two canonical variables of the corresponding classical Josephson Hamiltonian. We show that the adiabatic tunneling probabilities can be precisely predicted by the classical action at EPs in the weak nonreciprocal regime. In a certain region of strong nonreciprocity, we find that interestingly, the nonlinear interaction effects can be completely suppressed so that the adiabatic tunneling probabilities are identical to their linear counterparts. We also obtain a phase diagram for large ranges of nonreciprocity and nonlinear interaction parameters to explicitly demonstrate where the adiabaticity can break down, i.e., the emergence of the nonzero tunneling probabilities even in adiabatic limit.
We present a detailed analysis of the Landau–Zener problem for an interacting Bose–Einstein condensate in a time-varying double-well trap, especially focusing on the relation between the full many-particle problem and the mean-field approximation. Due to the nonlinear self-interaction a dynamical instability occurs, which leads to a breakdown of adiabaticity and thus fundamentally alters the dynamics. It is shown that essentially all the features of the Landau–Zener problem including the depletion of the condensate mode can be already understood within a semiclassical phase-space picture. In particular, this treatment resolves the formerly imputed incommutability of the adiabatic and semiclassical limits. The possibility of exploiting Landau–Zener sweeps to generate squeezed states for spectroscopic tasks is analyzed in detail. Moreover, we study the influence of phase noise and propose a Landau–Zener sweep as a sensitive yet readily implementable probe for decoherence, since the noise has significant effect on the transition rate for slow parameter variations.
Abstract Shortcuts to adiabaticity provide a general approach to mimic adiabatic quantum processes via arbitrarily fast evolutions in Hilbert space. For these counter-diabatic evolutions, higher speed comes at higher energy cost. Here, the counter-diabatic theory is employed as a minimal energy demanding scheme for speeding up adiabatic tasks. As a by-product, we show that this approach can be used to obtain infinite classes of transitionless models, including time-independent Hamiltonians under certain conditions over the eigenstates of the original Hamiltonian. We apply these results to investigate shortcuts to adiabaticity in decohering environments by introducing the requirement of a fixed energy resource. In this scenario, we show that generalized transitionless evolutions can be more robust against decoherence than their adiabatic counterparts. We illustrate this enhanced robustness both for the Landau–Zener model and for quantum gate Hamiltonians.
We consider dynamics of Dicke models, with and without counterrotating terms, under slow variations of parameters which drive the system through a quantum phase transition. The model without counterrotating terms and sweeped detuning is seen in the contexts of a many-body generalization of the Landau-Zener model and the dynamical passage through a second-order quantum phase transition (QPT). Adiabaticity is destroyed when the parameter crosses a critical value. Applying semiclassical analysis based on concepts of classical adiabatic invariants and mapping to the second Painleve equation (PII), we derive a formula which accurately describes particle distributions in the Hilbert space at wide range of parameters and initial conditions of the system. We find striking universal features in the particle distributions which can be probed in an experiment on Feshbach resonance passage or a cavity QED experiment. The dynamics is found to be crucially dependent on the direction of the sweep. The model with counterrotating terms has been realized recently in an experiment with ultracold atomic gases in a cavity. Its semiclassical dynamics is described by a Hamiltonian system with two degrees of freedom. Passage through a QPT corresponds to passage through a bifurcation, and can also be described by PII (after averaging over fast variables), leading to similar universal distributions. Under certain conditions, the Dicke model is reduced to the Lipkin-Meshkov-Glick model.