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arXiv:1803.05034 · Physical Review A

Pechukas-Yukawa formalism for Landau-Zener transitions in the presence of external noise

Mumnuna A. Qureshi, Johnny Q. Zhong, Peter Mason, Joseph J. Betouras, Alexandre M. Zagoskin

Published Jul 23, 2018• 3 citations
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Abstract

Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of one-dimensional classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearization. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.

DOI
10.1103/physreva.98.012128

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Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of one-dimensional classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearization. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.
Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of 1D classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearisation. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.
We analyze the influence of colored classical Gaussian noise on Landau-Zener transitions during a two-level crossing in a time-dependent regular external field. Transition probabilities and coherence factors become random due to the noise. We calculate their two-time correlation functions, which describe the response of this two-level system to a weak external pulse signal. The spectrum and intensity of the magnetic response are derived. Although the noise enters the equation of motion for the Bloch vector in a multiplicative way, nonperturbative analytic results are obtained by a resummation of diagrams in the limit of a short-noise correlation time. Our results also cover regimes where fluctuations are of the same order of magnitude as averages.
The degenerate Landau–Zener–Majorana–Stückelberg model consists of two degenerate energy levels whose energies vary with time and in the presence of an interaction which couples the states of the two levels. In the adiabatic limit, it allows for the populations transfer from states of one level to the states of the other level. The presence of an interaction with the environment influences the efficiency of the process. Nevertheless, identification of possible decoherence-free subspaces permits to engineer coupling schemes for which the effects of quantum noise can be made negligible.
We consider the evolution of the quantum states of a Hamiltonian that is parametrically perturbed via a term proportional to the adiabatic parameter $\ensuremath{\lambda}(t)$. Starting with the Pechukas-Yukawa mapping of the energy eigenvalue evolution in a generalized Calogero-Sutherland model of a one-dimensional classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of $d\ensuremath{\lambda}/dt$ and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of nonadiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfiability problem, we obtain the occupation dynamics, which provides insight into the population of states and sources of decoherence in a quantum system.
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of with cubic repulsion. Equilibrium statistical mechanics of this gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the Pechukas gas particles (i.e. energy levels in the system).
The effect of noise on the populations of the levels of a qubit in individual experimental implementations has been studied by the quantum trajectory method. A transition to the average dynamics obtained by means of multiple measurements of the state of the qubit is analyzed. The developed method is applied to investigate the effect of noise on the interference pattern appearing in the amplitude spectroscopy of the qubit in a strong variable field owing to Landau-Zener transitions. The effect of the number of repeated measurements and the fluctuation of the phase of a pump pulse on the formation of the response of the qubit to the external field has been analyzed. This makes it possible to interpret recent experiments in terms of individual implementations and averaged dynamics.