We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schr\"odinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem, which involves waiting at the minimum gap for a time ${t}_{w}$ ; we find an exact expression for the excitation probability as a function of ${t}_{w}$ . We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schr\"odinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem, which involves waiting at the minimum gap for a time ${t}_{w}$; we find an exact expression for the excitation probability as a function of ${t}_{w}$. We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.

arXiv:0910.5548Physical Review BFeb 19, 20109 citations

Adiabatic dynamics of many-body systems close to a quantum critical point

Tommaso Caneva

Oct 30, 20091 citation

Theory of anomalous Landau-Zener tunneling induced by nonlinear coupling

Wen-Yuan Wang, Hong-Juan Meng

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.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.

quant-pharXiv:2603.01523Physical Review AMar 3, 20260 citations

Adiabatic quenches through an extended quantum critical region

Franco Pellegrini, Simone Montangero, Giuseppe E. Santoro +1 more

By gradually changing the degree of the anisotropy in an $XXZ$ chain, we study the defect formation in a quantum system that crosses an extended critical region. We discuss two qualitatively different cases of quenches, from the antiferromagnetic to the ferromagnetic phase and from the critical to the antiferromagnetic phase. By means of time-dependent density matrix renormalization group simulations, we calculate the residual energy at the end of the quench as a characteristic quantity gauging the loss of adiabaticity. We find the dynamical scalings of the residual energy for both types of quenches, and compare them to the predictions of the Kibble-Zurek and Landau-Zener theories.By gradually changing the degree of the anisotropy in an $XXZ$ chain, we study the defect formation in a quantum system that crosses an extended critical region. We discuss two qualitatively different cases of quenches, from the antiferromagnetic to the ferromagnetic phase and from the critical to the antiferromagnetic phase. By means of time-dependent density matrix renormalization group simulations, we calculate the residual energy at the end of the quench as a characteristic quantity gauging the loss of adiabaticity. We find the dynamical scalings of the residual energy for both types of quenches, and compare them to the predictions of the Kibble-Zurek and Landau-Zener theories.

arXiv:0801.4475Physical Review BApr 30, 200864 citations

Non-Equilibrium Dynamics Across Quantum Critical Points: Slow Quenching

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti +3 more

Cambridge University Press eBooksJan 28, 20150 citations

{t}_{w}

; we find an exact expression for the excitation probability as a function of

{t}_{w}

. We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.

By gradually changing the degree of the anisotropy in an

XXZ

chain, we study the defect formation in a quantum system that crosses an extended critical region. We discuss two qualitatively different cases of quenches, from the antiferromagnetic to the ferromagnetic phase and from the critical to the antiferromagnetic phase. By means of time-dependent density matrix renormalization group simulations, we calculate the residual energy at the end of the quench as a characteristic quantity gauging the loss of adiabaticity. We find the dynamical scalings of the residual energy for both types of quenches, and compare them to the predictions of the Kibble-Zurek and Landau-Zener theories.