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arXiv:cond-mat/0511709 · Physical Review A

Adiabatic-impulse approximation for avoided level crossings: From phase-transition dynamics to Landau-Zener evolutions and back again

Bogdan Damski, Wojciech Hubert Zurek

Published Jun 8, 2006• 160 citations
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Abstract

We show that a simple approximation based on concepts underlying the Kibble-Zurek theory of second order phase-transition dynamics can be used to treat avoided level crossing problems. The approach discussed in this paper provides an intuitive insight into quantum dynamics of two-level systems, and may serve as a link between the theory of dynamics of classical and quantum phase transitions. To illustrate these ideas we analyze dynamics of a paramagnet-ferromagnet quantum phase transition in the Ising model. We also present exact unpublished solutions of the Landau-Zener-like problems.

DOI
10.1103/physreva.73.063405

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We show that a simple approximation based on concepts underlying the Kibble-Zurek theory of second order phase-transition dynamics can be used to treat avoided level crossing problems. The approach discussed in this paper provides an intuitive insight into quantum dynamics of two-level systems, and may serve as a link between the theory of dynamics of classical and quantum phase transitions. To illustrate these ideas we analyze dynamics of a paramagnet-ferromagnet quantum phase transition in the Ising model. We also present exact unpublished solutions of the Landau-Zener-like problems.
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We present two approaches to the dynamics of a quench-induced phase transition in the quantum Ising model. One follows the standard treatment of thermodynamic second order phase transitions but applies it to the quantum phase transitions. The other approach is quantum, and uses Landau-Zener formula for transition probabilities in avoided level crossings. We show that predictions of the two approaches of how the density of defects scales with the quench rate are compatible, and discuss the ensuing insights into the dynamics of quantum phase transitions.
The approach by Ettore Majorana for non-adiabatic transitions between two quasi-crossing levels is revisited and significantly extended. We rederive the transition probability, known as the Landau-Zener-Stückelberg-Majorana formula, and introduce Majorana's approach to modern readers. This result, typically referred as the Landau-Zener formula, was published by Majorana before Landau, Zener and Stückelberg. Moreover, we go well beyond previous results and we now obtain the full wave function, including its phase, which is important nowadays for quantum control and quantum information. The asymptotic wave function correctly describes the dynamics away from the avoided-level crossing, while it has limited accuracy in that region.
Ideally, in an adiabatic quantum computation or quantum state preparation, a simple ground state of an initial Hamiltonian is evolved adiabatically to an interesting ground state of a final Hamiltonian. Unfortunately, the simple and the interesting are often different enough to be separated by a quantum phase transition. Due to a vanishing energy gap between the ground state and excited states at the critical point, near the transition the adiabaticity is bound to fail. This failure is quantified by a quantum version of the Kibble-Zurek mechanism (KZM). In these notes I introduce KZM in its general textbook form, based on adiabatic/impulse approximation, and then support the picture by an exact solution of the integrable transverse field quantum Ising chain and an approximate one of the non-integrable Bose-Hubbard model. The last model illustrates typical problems with adiabatic quantum state preparation that are encoutered in atomic quantum simulators.
The Quantum Ising model is an exactly solvable model of quantum phase transition. This Letter gives an exact solution when the system is driven through the critical point at a finite rate. The evolution goes through a series of Landau-Zener level anticrossings when pairs of quasiparticles with opposite pseudomomenta get excited with a probability depending on the transition rate. The average density of defects excited in this way scales like a square root of the transition rate. This scaling is the same as the scaling obtained when the standard Kibble-Zurek mechanism of thermodynamic second order phase transitions is applied to the quantum phase transition in the Ising model.
We study the SSS matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].